Regularity-preserving elements of regular rings

INTRODUCTION

Variants of semigroups were first studied by Hickey¹, although variants of concrete semigroups of relations had earlier been considered by Magil^{2, 3}. We can see some properties of variants of semigroups in Refs. 1, 4, 5.

In this paper, we give the definition of variants of rings by using the concept of variants of semigroups and we characterize the regularity-preserving elements of regular rings.

REGULARITY-PRESERVING ELEMENTS OF REGULAR RINGS

Let R be a ring and a R. A new product ∘ is defined on R by x ∘ y = xay for all x,y R. Then (R,+,∘) is a ring. We usually write (R,+,a) rather that (R,+,∘) to make the element a explicit. The ring (R,+,a) is called a variant of R with respect to a.

An element a of a ring R is said to be regular if there exists x R such that a = axa. A ring R is called a regular ring if every element of R is regular.

Let R be a ring. An element a R is said to be a regularity-preserving element of R if the ring (R,+,a) is regular. Denote the set of all regularity-preserving elements of R by RP(R).

Proof : Assume that RP(R) is a nonempty set. Then there exists a R such that (R,+,a) is regular. Thus, for each x R, there exists y_x R such that x = x ∘ y_x ∘ x. Therefore, for all x R, x = x∘y_x ∘x = xay_xax = x(ay_xa)x. So x is regular in R. This implies that R is regular, a contradiction. □

Question   Let R be a regular ring. Is RP(R) a nonempty set?

The author has not been able to answer this question yet. However, the following theorem is true.

Proof : Let a,b RP(R) and x R. Then there exist y,z,s,t R such that x = xayax, x = xbzbx, a = absba, and b = batab. Thus

x	= xayax = x(absba)ya(xbzbx)
	= x(ab)(sbayaxbz)bx
	= x(ab)(sbayaxbz)(batab)x
	= x(ab)(sbayaxbzbat)(ab)x.

Therefore x is regular in (R,+,ab). Then ab RP(R). Hence RP(R) is a subsemigroup of (R,⋅). □

Now the author studies regularity-preserving elements of regular rings having an identity.

Let R be a ring with identity 1. An element a R is called a unit of R if there exist x,y R such that ax = 1 = ya (see Ref. 6). It is easy to prove that x = y. The following theorem holds.

Proof : Assume a is regularity-preserving. Then 1 is a regular element in (R,+,a), so there exists x R such that 1 = 1 ∘ x ∘ 1. Therefore 1 = 1 ∘ x ∘ 1 = 1axa1 = axa. Thus a is a unit of R.

Conversely, suppose that a is a unit of R. Let b R. Since R is regular, b = bxb for some x R, and so b = bxb = b1x1b = b(aa^-1)x(a^-1a)b = ba(a^-1xa^-1)ab. Therefore b is a regular element in (R,+,a). Hence a is a regularity-preserving element of R. □

The following corollary is obtained directly from Theorem 2 and Theorem 3.

Proof : It follows from Theorem 3 and the fact that the set of all units of R is a group under usual multiplication of R. □

Proof : Let a be a regularity-preserving element of R. Let b R. Then there exists x R such that b = b ∘ x ∘ b = baxab. Then b RaR. Therefore, RbR ⊆ RaR. □

The following two corollaries can be obtained directly from Theorem 4.

REGULARITY-PRESERVING ELEMENTS OF RINGS OF LINEAR TRANSFORMATIONS

Let V be a vector space over a field F and L(V ) be the set of all linear transformations on V . We know that (L(V ),+,∘) is a ring where ∘ is a composition of functions⁶. We have that the identity map on V is an identity of a ring L(V ). Moreover, L(V ) is a regular ring⁷. The following proposition is well-known.

By Theorem 3 and Proposition 1, the following corollary holds.

Let F be a field and M_n(F) denote the set of all n×n matrices on F . It is easy to prove that (M_n(F),+,⋅) is a ring where + and ⋅ is usual addition and usual multiplication of matrices, respectively. Moreover, the identity n × n matrix on F is an identity of a ring M_n(F). Let V be a vector space over F . If dimV = n, we know that a ring (M_n(F),+,⋅) is isomorphic to a ring (L(V ),+,∘)⁶. Therefore a ring M_n(F) is a regular ring. The following corollary follows from Corollary 4.

REGULARITY-PRESERVING ELEMENTS OF RINGS (ℤ_n,+,⋅)

Let ℤ and ℕ denote the set of all integers and the set of all positive integers, respectively. For n ℕ, let (ℤ_n,+,⋅) denote the ring of integers modulo n. For k ℤ, let be the equivalence class of integers modulo n containing k. We have that is an identity of a ring ℤ_n. The following proposition is well-known⁶.

Then the following corollary is true.

Proof : It follows from Theorem 1 and Proposition 3. □

Next, let n be a square-free number. By Proposition 3, the ring ℤ_n is regular.

Proof : It follows from Theorem 3, Proposition 2, and Proposition 3. □

Acknowledgements: The author would like to thank the referee for the helpful suggestions.

REFERENCES