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Abstract: We investigate the interaction between a ring R and the Cayley graph Cay((R)) of the semigroup of left ideals of R,as well as subdigraphs of this graph.Graph theoretic properties of these graphs are investigated,such as transitive closure,girth,radius,diameter,and spanning subgraphs.Conditions on certain of these graphs are given which imply that R is regular,left duo,or that the idempotents of R are central.We characterize simple rings in terms of Cay((R)).We characterize strongly regular rings in terms of a subdigraph of Cay((R)).
Key words: Cayley graph; semigroup; left ideal
CLC number: O157.5, O152.7, O153.3 Document code: A
Article ID: 1000-5137(2014)05-0506-05
2010 Mathematics Subject Classification. 05C25, 05E99, 20M10.
A directed graph or digraph G is a pair of sets,the set V of vertices and the set E of edges,denoted G=(V,E), where V≠ and EV×V.If v,w∈V and (v,w)∈E,then (v,w) is denoted by v w.The edge v w is said to be incident from v and incident to w.An edge from v to itself is a loop.We allow multiple edges between two vertices of G.Directed graphs with loops and multiple edges are often referred to as quivers.A subdigraph of G is a digraph G′=(V′,E′),where V′V and E′E.
A path of length n in G from vertex v to vertex w is a sequence of edges v v1 v2 … vn=w where the vertices are all distinct.The distance from v to w is the length of the shortest path from v to w.The eccentricity of a vertex v is the maximum of its finite distances to all other vertices.The radius of G is the minimum eccentricity of all its vertices.The diameter of G is the maximum eccentricity of all its vertices; i.e.,the diameter is the maximum distance between all pairs of distinct vertices in G.
A digraph G is strongly connected if there is a path from any vertex to any other vertex.A maximal strongly connected subdigraph of G is a strongly connected component of G.
A cycle from vertex v to itself in G is a path from v to some vertex w together with an edge from w to v.The girth of G is the length of the smallest cycle in G.The circumference of G is the length of the largest cycle in G.
If G=(V,E) is a directed graph,then the underlying graph of G is the undirected graph G′= (V′,E′ ),where V′=V and E′={{v,w}|(v,w)∈E}.That is,G′ is constructed by taking G and "removing the directions" on the edges.As before,V′ is the set of vertices of G′,and E′ is the set of edges of G′.The edge {v,w} is denoted v-w.A digraph G is called weakly connected if its underlying graph is a connected graph. Let S be a semigroup.The Cayley graph of S,denoted Cay(S),is the digraph G=(S,E) where E={(s,t)∈S×S|sx=t for some x∈S}.An edge (s,t) is denoted by s x t if sx=t,and the edge is said to be labeled by x.Let ≠XS.Define the subdigraph Cay(S,X)=(S,EX),where EX is the set of all edges labeled by the elements of X.
Given a Cayley graph,we can define the underlying non-directed graph as above by replacing multiple edges by a single edge and removing the directions on the edges.We also remove the labels on the edges.We do not eliminate loops.
Let R be a ring with identity 1≠0 which is not necessarily commutative.The set of left ideals (R) of R forms a semigroup under left ideal multiplication.Therefore,we can define the Cayley graph Cay((R)),as well as subdigraphs of this graph.As above,if ≠X(R),then Cay((R),X ) is the subdigraph of Cay((R)) whose edges are labeled by the elements of X.If we assume that R is a prime ring,then the set *(R) of nonzero left ideals of R forms a semigroup under left ideal multiplication,so we can also define the Cayley graph Cay(*(R)).Again,if X*(R),then Cay(*(R),X ) is the subdigraph of Cay(*(R))
The Cayley graph built upon the semigroup of left ideals of a ring 2 Basic Graph-Theoretic Properties
Definition 1 Let G be a digraph.The transitive closure of G is the graph obtained by adding an edge v w whenever there is a path from v to w.If a digraph G is equal to its transitive closure,then G is said to be transitively closed.
Proposition 2 Cay((R)) is transitively closed.
Proof Let H,K be any two left ideals of R.Suppose that there is a path from H to K of the formH A1 X1 A2 … An-1 Xn-1 An K.Then A=∏nj=1Aj is a left ideal of R,and there is an edge H A K.
Example 3 The Calyley graph needs not be connected.Let R=
Proposition 4 The underlying graph of Cay((R)) has diameter at most 2.
Proof There are edges H 0 0 and K 0 0 in the Cayley graph,which yields a path H-0-K in the underlying graph.
Proposition 5 Suppose there is a directed path of length n from H to K in Cay((R)).
1.If n≥2,then in the underlying graph the corresponding undirected path is in a cycle of length n+1.
2.Let n≥3.Fix an edge e in the path from H to K.Then there is a path from H to K of length 3 which includes the edge e.
3.For every 1≤j＜n there is a directed path of length j from H to K in Cay((R)).
Proposition 6 Let U denote the underlying graph of Cay((R)).Let gir(U) and cir(U) denote the girth and circumference of U,respectively.Let rad and diam denote the radius and diameter,respectively,of Cay((R)).Then gir(U)≤rad(Cay((R)))+1≤diam(Cay((R)))+1≤cir(U). Proof Suppose that rad(Cay((R)))＜gir(U)-1.Then there is a path in the digraph u … v of length rad(Cay((R)))＜gir(U)-1.By Proposition 2 the digraph contains an edge from u to v.This yields a cycle in the underlying graph of length rad(Cay((R)))+1.Hence there is a cycle in the underlying graph of length rad(Cay((R)))+1＜gir(U),contrary to the definition of girth.Hence rad(Cay((R)))≥gir(U)-1.
By definition rad(Cay((R)))≤diam(Cay((R))),which impliesrad(Cay((R)))+1≤diam(Cay((R)))+1. Finally,we can repeat the argument in the first paragraph to show that diam(Cay((R)))+1≤cir(U).
3 Stars and Completeness
Definition 7 A directed graph G without cycles is an inward star if there is a vertex v such that every vertex is adjacent to v and every edge is labeled by v.In this case we say that v is the center of G.
Definition 8 Let G be a directed graph.A subdigraph G′ of G is a spanning subdigraph of G if G′ contains all the vertices of G.
Definition 9 A ring R is subdirectly irreducible if the intersection A of all the non-zero two-sided ideals of R is non-zero.The ideal A is the heart of R.
Theorem 10 Let R be prime.If R is subdirectly irreducible with heart A then Cay(*(R)) contains a spanning subdigraph which is an inward star centered at A and whose edges are all labeled by A.
Conversely,suppose that Cay(*(R)) contains a spanning subdigraph which is an inward star centered at a two-sided ideal A≠0 and whose edges are all labeled by A.Then R is subdirectly irreducible with heart A.
Proof Let R be prime,and suppose that R is subdirectly irreducible with heart A.For any non-zero left ideal H we have that HA is a non-zero two-sided ideal of R contained in A,which implies that HA=A.Thus H AA.
Conversely,suppose that R is prime and that Cay(*(R)) contains a spanning subgraph which is an inward star centered at a two-sided ideal A≠0 and whose edges are all labeled by A.Let I≠0 be a two-sided ideal of R.Then 0≠A=IAI.Hence the non-zero two-sided ideals of R have a non-zero intersection containing A,so that R is subdirectly irreducible,say with heart H.Then AH.But A is a non-zero two-sided ideal,so that HA.Therefore A=H.
We can apply this result to simple rings.The next result is from ［1,Prop.4.3］.
Lemma 11 A ring R is simple if and only if HK=K for any non-zero left ideals H,K.
Proof Let R be simple and let H,K be non-zero left ideals of R.Then HK=H(RK)=(HR)K=RK=K.Conversely,if HK=K for any non-zero left ideals H,K of R,then let H=Ra for any 0≠a∈R,and let K=R.Then RaR=R for any 0≠a∈R,and hence R is simple. Definition 12 A directed graph is complete if between any two vertices there is an edge in each direction.
Theorem 13 A ring R is simple if and only if Cay(*(R)),*(R)) is a complete digraph which also has a loop on each vertex.
Proof Let R be a simple ring.Then HK=K for every pair of nonzero left ideals H,K,by Lemma 11.Therefore,the graph contains the edge H K K for any pair of vertices H,K.If there is another edge H XK, then HX=K.But by Lemma 11 we have that HX=X.Thus X=K and the edge from H to K is unique.Finally,when H=K we have a loop H H H.
Conversely,suppose that Cay(*(R),*(R)) is a complete digraph which also has a loop on each vertex.Let 0≠a∈R.Then there is an edge Ra X R for some left ideal X.Therefore R=RaXX,which implies that R=X.This means that R=RaR.Since a is arbitrary,we have that R is simple.
Corollary 14 A ring R is simple if and only if there is a unique edge between any two vertices H,K in Cay(*(R),*(R)),namely H K K.
4 Cycles and Loops
In this section we describe connections between loops in Cay((R)) and regularity conditions in R.Many of these results appear in different forms in ［2］.
Proposition 16 If Cay((R)) has no cycles except possibly for loops,then every idempotent in R is central.
Proof This proof is from ［2,Prop.4.1］.Let 0≠e=e2∈R.Then ReR=Re?R and Re=ReRe=ReR?Re .Therefore,there is a cycle Re R ReR Re Re.By hypothesis we have that Re=ReR.In particular,ReeR.Hence,for any r∈R there is x∈R such that er=xe,and so ere=xee=xe=er.(Such idempotents are called right semicentral; see ［3］.)
Let f be any other idempotent in R.Then ef-fe=1?(ef-fe)=(f+1-f) (ef-fe).Now f(ef-fe)=fef-fe=fef-fef=0.Also,(1-f)(ef-fe)= (1-f)(ef-fe)(1-f).A straightforward computation shows that this expression is 0.Therefore ef=fe for any two idempotents.It is known that,in any ring,if all the idempotents commute with each other,then the idempotents are central.
Definition 17 A semigroup S with 0 is right cancellative if xs=ys≠0 implies x=y for all 0≠s,x,y∈S.
Proposition 18 Suppose that the maps in Cay((R)) induced by any left ideal are one-to-one; that is,if there are edges X H K and Y H K,then X=Y.Then R is left duo and the idempotents of R are all central.
Proof Suppose that XH=YH= K for left ideals H,X,Y,K.Then there are two edges X H K and Y H K.By hypothesis X=Y.Therefore the semigroup (R) is right cancellative.
If H is a left ideal of R,then HR=HR2.By right cancellation,we have H=HR.Hence R is left duo.In particular,if e is an idempotent of R,then Re=ReR.The result now follows from the proof of Proposition 16. Note that the power series ring in one indeterminate over a field satisfies the hypothesis of Proposition 16.Commutativity of the idempotents under the hypotheses of Proposition 16 also follows from ［2,Prop.2.8 and Prop.4.1］.
Definition 19 A ring R is left weakly regular (l.w.r.) if H2=H for any left ideal H of R.Right weakly regular rings are defined dually.
For a survey on right weakly regular rings,see ［4］.
Lemma 20 ［2,Prop.3.1(a)］ If M is a maximal left ideal of R,then either M2=M or M is a two-sided ideal.
Proposition 21 If every vertex H in Cay((R)) has a loop H H H,and if there are no edges going into the vertex labeled by R,then R is regular.
Proof Let Cay((R)) satisfy the conditions above.The existence of loops H H H means that H2=H,and hence R is (l.w.r.).
Suppose that the graph has no edges going to the vertex labeled by R.Let M be a maximal left ideal.By Lemma 20 either M is a two-sided ideal or MR=R.This latter condition is equivalent to the existence of an edge M R R.Since this case cannot occur by hypothesis,it follows that every maximal left ideal of R is two-sided.From ［5,Thm.2.7］ if R is (l.w.r.) and every maximal left ideal is two-sided,then R is a regular ring.
The converse of Proposition 21 is false.Let R be a simple ring.Then R is (l.w.r.) by Lemma 11,but HR=R for avery nonzero left ideal H of R.
Corollary 22 Let R satisfy the conditions of Proposition 21.Let J(R) denote the Jacobson radical of R.Then R/J(R) is a subdirect product of simple rings.
Proposition 23 A ring R is strongly regular if and only if every vertex H in Cay((R)) has a loop H H H and Cay((R)) has no other cycles.
Proof Let R be strongly regular.Then every left ideal H is generated by central idempotents,so that H2=H and H is a two-sided ideal.The graph then has a loop H H H.
Suppose that the ideals H,K are in the same cycle.By Prop.2 there exist ideals X,Y such that HX=K and KX=H.But then HK and KH,so that H=K.
Conversely,let Cay((R)) satisfy the conditions above.The existence of loops H H H means that R is (l.w.r.).
Let M be a maximal left ideal of R.If M is not two-sided,then there is an edge M R R.However,RM=M.Thus there is a cycle MRRMM which implies that M=R,contradiction.It follows that MR=M.As before,since R is (l.w.r.) and every maximal left ideal of R is two-sided,we have that R is regular by ［5,Thm.2.7］.Since there are no cycles in Cay((R)) except for loops,then by Proposition 16 all idempotents of R are central.Therefore R is strongly regular.
References:
［1］ H.E.Heatherly,R.P.Tucci.The Semigroup of Right Ideals of a Ring ［J］.Math.Pannonica,2007,18/1:19-26.
［2］ H.E.Heatherly,R.P.Tucci.Rings Whose Semigroup of Right Ideals is J-Trivial ［J］.The Intrenational Electronic Journal of Algebra,2011,10:151-161.
［3］ G.F.Birkenmeier.Idempotents and Completely Semiprime Ideals ［J］.Comm.Algebra,1983,11:567-580.
［4］ H.E.Heatherly,R.P.Tucci.Right Weakly Regular Rings:A Survey, ［C］// T.Albu,G.F.Birkenmeier,A.Erdo gan,A.Tercan.Ring and Module Theory.Basel:Springer Verlag Trends,2010,115-124.
［5］ H.P.Yu.On Quasi-Duo Rings ［J］.Glasgow Math.J.,1995,37:21-31.
(Zhenzhen Feng)
Key words: Cayley graph; semigroup; left ideal
CLC number: O157.5, O152.7, O153.3 Document code: A
Article ID: 1000-5137(2014)05-0506-05
2010 Mathematics Subject Classification. 05C25, 05E99, 20M10.
A directed graph or digraph G is a pair of sets,the set V of vertices and the set E of edges,denoted G=(V,E), where V≠ and EV×V.If v,w∈V and (v,w)∈E,then (v,w) is denoted by v w.The edge v w is said to be incident from v and incident to w.An edge from v to itself is a loop.We allow multiple edges between two vertices of G.Directed graphs with loops and multiple edges are often referred to as quivers.A subdigraph of G is a digraph G′=(V′,E′),where V′V and E′E.
A path of length n in G from vertex v to vertex w is a sequence of edges v v1 v2 … vn=w where the vertices are all distinct.The distance from v to w is the length of the shortest path from v to w.The eccentricity of a vertex v is the maximum of its finite distances to all other vertices.The radius of G is the minimum eccentricity of all its vertices.The diameter of G is the maximum eccentricity of all its vertices; i.e.,the diameter is the maximum distance between all pairs of distinct vertices in G.
A digraph G is strongly connected if there is a path from any vertex to any other vertex.A maximal strongly connected subdigraph of G is a strongly connected component of G.
A cycle from vertex v to itself in G is a path from v to some vertex w together with an edge from w to v.The girth of G is the length of the smallest cycle in G.The circumference of G is the length of the largest cycle in G.
If G=(V,E) is a directed graph,then the underlying graph of G is the undirected graph G′= (V′,E′ ),where V′=V and E′={{v,w}|(v,w)∈E}.That is,G′ is constructed by taking G and "removing the directions" on the edges.As before,V′ is the set of vertices of G′,and E′ is the set of edges of G′.The edge {v,w} is denoted v-w.A digraph G is called weakly connected if its underlying graph is a connected graph. Let S be a semigroup.The Cayley graph of S,denoted Cay(S),is the digraph G=(S,E) where E={(s,t)∈S×S|sx=t for some x∈S}.An edge (s,t) is denoted by s x t if sx=t,and the edge is said to be labeled by x.Let ≠XS.Define the subdigraph Cay(S,X)=(S,EX),where EX is the set of all edges labeled by the elements of X.
Given a Cayley graph,we can define the underlying non-directed graph as above by replacing multiple edges by a single edge and removing the directions on the edges.We also remove the labels on the edges.We do not eliminate loops.
Let R be a ring with identity 1≠0 which is not necessarily commutative.The set of left ideals (R) of R forms a semigroup under left ideal multiplication.Therefore,we can define the Cayley graph Cay((R)),as well as subdigraphs of this graph.As above,if ≠X(R),then Cay((R),X ) is the subdigraph of Cay((R)) whose edges are labeled by the elements of X.If we assume that R is a prime ring,then the set *(R) of nonzero left ideals of R forms a semigroup under left ideal multiplication,so we can also define the Cayley graph Cay(*(R)).Again,if X*(R),then Cay(*(R),X ) is the subdigraph of Cay(*(R))
The Cayley graph built upon the semigroup of left ideals of a ring 2 Basic Graph-Theoretic Properties
Definition 1 Let G be a digraph.The transitive closure of G is the graph obtained by adding an edge v w whenever there is a path from v to w.If a digraph G is equal to its transitive closure,then G is said to be transitively closed.
Proposition 2 Cay((R)) is transitively closed.
Proof Let H,K be any two left ideals of R.Suppose that there is a path from H to K of the formH A1 X1 A2 … An-1 Xn-1 An K.Then A=∏nj=1Aj is a left ideal of R,and there is an edge H A K.
Example 3 The Calyley graph needs not be connected.Let R=
Proposition 4 The underlying graph of Cay((R)) has diameter at most 2.
Proof There are edges H 0 0 and K 0 0 in the Cayley graph,which yields a path H-0-K in the underlying graph.
Proposition 5 Suppose there is a directed path of length n from H to K in Cay((R)).
1.If n≥2,then in the underlying graph the corresponding undirected path is in a cycle of length n+1.
2.Let n≥3.Fix an edge e in the path from H to K.Then there is a path from H to K of length 3 which includes the edge e.
3.For every 1≤j＜n there is a directed path of length j from H to K in Cay((R)).
Proposition 6 Let U denote the underlying graph of Cay((R)).Let gir(U) and cir(U) denote the girth and circumference of U,respectively.Let rad and diam denote the radius and diameter,respectively,of Cay((R)).Then gir(U)≤rad(Cay((R)))+1≤diam(Cay((R)))+1≤cir(U). Proof Suppose that rad(Cay((R)))＜gir(U)-1.Then there is a path in the digraph u … v of length rad(Cay((R)))＜gir(U)-1.By Proposition 2 the digraph contains an edge from u to v.This yields a cycle in the underlying graph of length rad(Cay((R)))+1.Hence there is a cycle in the underlying graph of length rad(Cay((R)))+1＜gir(U),contrary to the definition of girth.Hence rad(Cay((R)))≥gir(U)-1.
By definition rad(Cay((R)))≤diam(Cay((R))),which impliesrad(Cay((R)))+1≤diam(Cay((R)))+1. Finally,we can repeat the argument in the first paragraph to show that diam(Cay((R)))+1≤cir(U).
3 Stars and Completeness
Definition 7 A directed graph G without cycles is an inward star if there is a vertex v such that every vertex is adjacent to v and every edge is labeled by v.In this case we say that v is the center of G.
Definition 8 Let G be a directed graph.A subdigraph G′ of G is a spanning subdigraph of G if G′ contains all the vertices of G.
Definition 9 A ring R is subdirectly irreducible if the intersection A of all the non-zero two-sided ideals of R is non-zero.The ideal A is the heart of R.
Theorem 10 Let R be prime.If R is subdirectly irreducible with heart A then Cay(*(R)) contains a spanning subdigraph which is an inward star centered at A and whose edges are all labeled by A.
Conversely,suppose that Cay(*(R)) contains a spanning subdigraph which is an inward star centered at a two-sided ideal A≠0 and whose edges are all labeled by A.Then R is subdirectly irreducible with heart A.
Proof Let R be prime,and suppose that R is subdirectly irreducible with heart A.For any non-zero left ideal H we have that HA is a non-zero two-sided ideal of R contained in A,which implies that HA=A.Thus H AA.
Conversely,suppose that R is prime and that Cay(*(R)) contains a spanning subgraph which is an inward star centered at a two-sided ideal A≠0 and whose edges are all labeled by A.Let I≠0 be a two-sided ideal of R.Then 0≠A=IAI.Hence the non-zero two-sided ideals of R have a non-zero intersection containing A,so that R is subdirectly irreducible,say with heart H.Then AH.But A is a non-zero two-sided ideal,so that HA.Therefore A=H.
We can apply this result to simple rings.The next result is from ［1,Prop.4.3］.
Lemma 11 A ring R is simple if and only if HK=K for any non-zero left ideals H,K.
Proof Let R be simple and let H,K be non-zero left ideals of R.Then HK=H(RK)=(HR)K=RK=K.Conversely,if HK=K for any non-zero left ideals H,K of R,then let H=Ra for any 0≠a∈R,and let K=R.Then RaR=R for any 0≠a∈R,and hence R is simple. Definition 12 A directed graph is complete if between any two vertices there is an edge in each direction.
Theorem 13 A ring R is simple if and only if Cay(*(R)),*(R)) is a complete digraph which also has a loop on each vertex.
Proof Let R be a simple ring.Then HK=K for every pair of nonzero left ideals H,K,by Lemma 11.Therefore,the graph contains the edge H K K for any pair of vertices H,K.If there is another edge H XK, then HX=K.But by Lemma 11 we have that HX=X.Thus X=K and the edge from H to K is unique.Finally,when H=K we have a loop H H H.
Conversely,suppose that Cay(*(R),*(R)) is a complete digraph which also has a loop on each vertex.Let 0≠a∈R.Then there is an edge Ra X R for some left ideal X.Therefore R=RaXX,which implies that R=X.This means that R=RaR.Since a is arbitrary,we have that R is simple.
Corollary 14 A ring R is simple if and only if there is a unique edge between any two vertices H,K in Cay(*(R),*(R)),namely H K K.
4 Cycles and Loops
In this section we describe connections between loops in Cay((R)) and regularity conditions in R.Many of these results appear in different forms in ［2］.
Proposition 16 If Cay((R)) has no cycles except possibly for loops,then every idempotent in R is central.
Proof This proof is from ［2,Prop.4.1］.Let 0≠e=e2∈R.Then ReR=Re?R and Re=ReRe=ReR?Re .Therefore,there is a cycle Re R ReR Re Re.By hypothesis we have that Re=ReR.In particular,ReeR.Hence,for any r∈R there is x∈R such that er=xe,and so ere=xee=xe=er.(Such idempotents are called right semicentral; see ［3］.)
Let f be any other idempotent in R.Then ef-fe=1?(ef-fe)=(f+1-f) (ef-fe).Now f(ef-fe)=fef-fe=fef-fef=0.Also,(1-f)(ef-fe)= (1-f)(ef-fe)(1-f).A straightforward computation shows that this expression is 0.Therefore ef=fe for any two idempotents.It is known that,in any ring,if all the idempotents commute with each other,then the idempotents are central.
Definition 17 A semigroup S with 0 is right cancellative if xs=ys≠0 implies x=y for all 0≠s,x,y∈S.
Proposition 18 Suppose that the maps in Cay((R)) induced by any left ideal are one-to-one; that is,if there are edges X H K and Y H K,then X=Y.Then R is left duo and the idempotents of R are all central.
Proof Suppose that XH=YH= K for left ideals H,X,Y,K.Then there are two edges X H K and Y H K.By hypothesis X=Y.Therefore the semigroup (R) is right cancellative.
If H is a left ideal of R,then HR=HR2.By right cancellation,we have H=HR.Hence R is left duo.In particular,if e is an idempotent of R,then Re=ReR.The result now follows from the proof of Proposition 16. Note that the power series ring in one indeterminate over a field satisfies the hypothesis of Proposition 16.Commutativity of the idempotents under the hypotheses of Proposition 16 also follows from ［2,Prop.2.8 and Prop.4.1］.
Definition 19 A ring R is left weakly regular (l.w.r.) if H2=H for any left ideal H of R.Right weakly regular rings are defined dually.
For a survey on right weakly regular rings,see ［4］.
Lemma 20 ［2,Prop.3.1(a)］ If M is a maximal left ideal of R,then either M2=M or M is a two-sided ideal.
Proposition 21 If every vertex H in Cay((R)) has a loop H H H,and if there are no edges going into the vertex labeled by R,then R is regular.
Proof Let Cay((R)) satisfy the conditions above.The existence of loops H H H means that H2=H,and hence R is (l.w.r.).
Suppose that the graph has no edges going to the vertex labeled by R.Let M be a maximal left ideal.By Lemma 20 either M is a two-sided ideal or MR=R.This latter condition is equivalent to the existence of an edge M R R.Since this case cannot occur by hypothesis,it follows that every maximal left ideal of R is two-sided.From ［5,Thm.2.7］ if R is (l.w.r.) and every maximal left ideal is two-sided,then R is a regular ring.
The converse of Proposition 21 is false.Let R be a simple ring.Then R is (l.w.r.) by Lemma 11,but HR=R for avery nonzero left ideal H of R.
Corollary 22 Let R satisfy the conditions of Proposition 21.Let J(R) denote the Jacobson radical of R.Then R/J(R) is a subdirect product of simple rings.
Proposition 23 A ring R is strongly regular if and only if every vertex H in Cay((R)) has a loop H H H and Cay((R)) has no other cycles.
Proof Let R be strongly regular.Then every left ideal H is generated by central idempotents,so that H2=H and H is a two-sided ideal.The graph then has a loop H H H.
Suppose that the ideals H,K are in the same cycle.By Prop.2 there exist ideals X,Y such that HX=K and KX=H.But then HK and KH,so that H=K.
Conversely,let Cay((R)) satisfy the conditions above.The existence of loops H H H means that R is (l.w.r.).
Let M be a maximal left ideal of R.If M is not two-sided,then there is an edge M R R.However,RM=M.Thus there is a cycle MRRMM which implies that M=R,contradiction.It follows that MR=M.As before,since R is (l.w.r.) and every maximal left ideal of R is two-sided,we have that R is regular by ［5,Thm.2.7］.Since there are no cycles in Cay((R)) except for loops,then by Proposition 16 all idempotents of R are central.Therefore R is strongly regular.
References:
［1］ H.E.Heatherly,R.P.Tucci.The Semigroup of Right Ideals of a Ring ［J］.Math.Pannonica,2007,18/1:19-26.
［2］ H.E.Heatherly,R.P.Tucci.Rings Whose Semigroup of Right Ideals is J-Trivial ［J］.The Intrenational Electronic Journal of Algebra,2011,10:151-161.
［3］ G.F.Birkenmeier.Idempotents and Completely Semiprime Ideals ［J］.Comm.Algebra,1983,11:567-580.
［4］ H.E.Heatherly,R.P.Tucci.Right Weakly Regular Rings:A Survey, ［C］// T.Albu,G.F.Birkenmeier,A.Erdo gan,A.Tercan.Ring and Module Theory.Basel:Springer Verlag Trends,2010,115-124.
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(Zhenzhen Feng)