aa r X i v : . [ m a t h . R A ] O c t A PARTITION OF FINITE RINGS
VINEETH CHINTALA
University of Hyderabad, Hyderabad 500046 Telangana, India.
Abstract.
Given any finite ring, we’ll construct a partition of it, where eachblock corresponds to one idempotent. The partition is so simple that it worksnot only for associative algebras but also for (finite) power-associative algebras.We prove in particular that idempotents and regular elements can always belifted (over homomorphisms) for finite rings.
Keywords.
Partition; Idempo-tent; Power-associative algebra. Idempotent Partitions
Given any finite ring, we’ll construct a partition of it where each block corre-sponds to one idempotent. The partition is so simple that it works for any finitepower-associative algebra.An element e is said to be an idempotent if e = e . Idempotents play a centralrole in the classification of algebras, and the partition indicates that they mayalso have some combinatorial significance. We’ll use the partition to show thatidempotents can always be lifted for finite rings (Theorem 1 . Definition 1.1.
A ring R is said to be power-associative if the subring generatedby any one element is associative. Concretely, this means that if x ∈ R then thereis no ambiguity in the product x n . For example x · x is the same as x · x . Power-associativity is one of the simplest restrictions that one can place on a ring.It is satisfied by many useful rings including all associative, Jordan and the Cayley-Dickson algebras. Throughout the paper, we’ll assume that the ring R is power-associative and has an identity element. Here’s an interesting property of finiterings. Lemma 1.2.
Let R be a finite power-associative ring and x ∈ R .(1) Then e x = x n is an idempotent for some positive integer n .(2) Further, the idempotent e x is uniquely determined by x ∈ R .Proof. The set { x i : i ≥ } is finite and so x r = ( x r ) t for some integers r, t > y = x r . Then y t − is an idempotent. Indeed, y t − = y t − y t = y t − y = y t − . To prove uniqueness, suppose x r = e and x s = e . Then e = ( x r ) s = ( x s ) r = e . (cid:3) E-mail address : [email protected] .The author is supported by the DST-Inspire faculty fellowship in India. Definition 1.3.
For each idempotent e ∈ R , define B e = { x ∈ R : x k = e for some positive integer k } . Put every element x in the corresponding block B e x . This gives a partition ofthe ring into blocks, where each block corresponds to one idempotent. R = B ⊔ B ⊔ · · · Here we’re talking about partitions of rings as sets. For such partitions to be use-ful, they should be compatible with ring homomorphisms. The following propertyis easy to check.
Theorem 1.4.
Let φ be a ring homomorphism R → R ′ . The map φ preserves theiridempotent partitions R R ′ { B e } { B ′ e ′ } φφ where { B e } and { B ′ e ′ } denote the idempotent-partitions of R, R ′ respectively.Proof. Let a ∈ B e . Note that φ ( a n ) = φ ( a ) n for all positive integers n . Therefore φ ( a ) ∈ B ′ φ ( e ) . In other words φ ( B e ) ⊆ B ′ φ ( e ) . In fact, when φ is surjective we have B ′ e ′ = { [ e φ ( B e ) : φ ( e ) = e ′ } . (cid:3) Notice that the number of blocks in φ ( R ) is at most the number of blocks in R . For any ring R , let idem ( R ) denote the set of its idempotents. Given any map φ : R → S , we say that idempotents can be lifted (over φ ) if every idempotent in S has a pre-image in R which is also an idempotent. Theorem 1.5.
Let φ : R → S be a surjective homomorphism between two finitepower-associative algebras. Then(1) Every idempotent of S can be lifted to an idempotent in R .(2) | idem ( R ) | ≥ | idem ( S ) | .Proof. Let e ′ ∈ S be an idempotent. Since φ is surjective, there exists a ∈ R suchthat φ ( a ) = e ′ . Then φ ( e a ) = e ′ where a ∈ B e a . The second part follows from thefirst, as distinct idempotents of S lift to distinct idempotents of R . (cid:3) The above theorem is not obvious from the homomorphism itself as there canbe many elements x ∈ R such that x − x ∈ ker ( φ ), making φ ( x ) an idempotent.The above theorem says that essentially for finite rings, no new idempotents aregenerated after a homomorphism.For associative rings, it is known that idempotents in R/I lift to R when I isa nil ideal or when R is I - adically complete ([5], Theorems 21.28, 21.31). Neitherof these conditions hold when I contains an idempotent. The paper [2] gives a fewcounterexamples where idempotents cannot always be lifted in infinite associativerings. PARTITION OF FINITE RINGS 3 Finite associative rings
In the remainder of the paper, we’ll work with only finite associative rings.
Theorem 2.1.
Let R be a finite associative ring, and I be a left (or right) ideal.Then idempotents in R/I can be lifted to idempotents in R .Proof. The proof is the same as in Theorem 1 .
5. Let a ∈ R be such that a ∈ R/I is an idempotent. Since R is a finite ring, we know that e a = a n is an idempotentfor some positive integer n . Clearly e a = a . (cid:3) The lifting of idempotents opens the door for the lifting of other properties.(See [4] for some consequences of idempotent lifting). We’ll now see that regular elements can also be lifted. An element x is said to be regular (or von Neumannregular ) if xyx = x for some element y ∈ R . Notice that every idempotent e isregular as e = e (take x = e, y = 1).In general, lifting of idempotents does not imply lifting of regular elements. Fora counterexample, consider the map φ : Z → Z / Z . Here Z / Z has only trivialidempotents { ¯0 , ¯1 } which can obviously be lifted, but some of its regular elements { , } cannot be lifted to regular elements in Z . However, it was shown that ifidempotents lift modulo every left ideal contained in a two-sided ideal I , thenregular elements lift modulo I ([3], Theorem 9.3). Unsurprisingly the proof is a bitsimpler for finite rings.The following result shows that regular elements can always be lifted for finiterings. Theorem 2.2.
Let R be a finite associative ring and I be a left ideal. Then regularelements in R/I can be lifted to regular elements in R .Proof. Let x, y be two elements such that xyx − x ∈ I . Since R is a finite ring, e = ( xy ) n is an idempotent for some positive integer n . We need to show that thereis a regular element z ∈ R such that z − x ∈ I .Take z = ( xy ) n − x . Then zy = e and ez = z . Therefore zyz = ez = z. Further z − x = (( xy ) n − − x = r ( xy − x for some r ∈ R . Since ( xy − x = xyx − x ∈ I , we have z − x ∈ I . (cid:3) Zero Divisors in associative rings.Theorem 2.4.
Let R be a finite associative ring with partition { B e : e ∈ Idem ( R ) } .The blocks B e satisfy the following properties.(1) Let a i ∈ B e i . If a a = 0 , then e e = 0 .(2) Let a, b ∈ B e . If ab = 0 , then e = 0 .(3) B consists of all nilpotent elements and B consists of all invertible ele-ments in R .Proof. Suppose a r = e and a s = e . Then a a = 0 implies that e e = a r a s = 0 . A PARTITION OF FINITE RINGS
Taking e = e , the second statement obviously follows.If an element a is invertible, then so is e a . Since e a = e a it follows that e a = 1.Finally, note that any element x ∈ B satisfies x n = 0 for some positive integer n . (cid:3) Since e x = e x , the element e x is a zero divisor if e x = 1. In that case x will alsobe a zero divisor. Therefore { B e | e = 1 } is a partition of the zero-divisors of R . Definition 2.5.
Following [1] one can consider any subset S ⊆ R as a directedgraph, where there is an edge a −→ b between two elements a, b if and only if ab = 0 . We’ll refer to this graph as the zero-divisor graph Γ( S ) . Suppose there is an edge a −→ b between two elements a, b ∈ Γ( R ). Then itfollows (from Theorem 2 .
4) that there is also an edge e a −→ e b . In particular, if e = 0 then there are no edges between elements inside B e . Theorem 2.6.
Let R be a finite associative ring. Then { B e : e = 0 } is a partitionof the subgraph Γ( R \ B ) . A subset { x , · · · , x n } is called a clique if x i −→ x j and x j −→ x i for all i = j . Theorem 2.7. If { e , · · · , e n } is a maximal clique of non-zero idempotents in Γ( R ) , then n P i =1 e i = 1 . Proof.
Let d = 1 − n P i =1 e i . Then d is also an idempotent and de i = e i d = 0.Therefore { d, e , · · · , e n } is a larger clique unless d = 0. (cid:3) References [1] I. Beck, Coloring of commutative rings,
Journal of Algebra (1988), 116 (1): 208–226.[2] Alexander J. Diesl, Samuel J. Dittmer, and Pace P. Nielsen: Idempotent lifting and ringextensions.
J. Algebra Appl.
J. Pure Appl. Algebra