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* Corresponding author, E-mail: email@example.com
Received 13 May 2008, Accepted 14 Jan 2009
REGULARITY-PRESERVING ELEMENTS OF REGULAR RINGS
REGULARITY-PRESERVING ELEMENTS OF RINGS OF LINEAR TRANSFORMATIONS
REGULARITY-PRESERVING ELEMENTS OF RINGS (ℤn,+,⋅)
Variants of semigroups were first studied by Hickey1, although variants of concrete semigroups of relations had earlier been considered by Magil2, 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.
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 yx R such that x = x ∘ yx ∘ x. Therefore, for all x R, x = x∘yx ∘x = xayxax = x(ayxa)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
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. □
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.
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.
Let F be a field and Mn(F) denote the set of all n×n matrices on F . It is easy to prove that (Mn(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 Mn(F). Let V be a vector space over F . If dimV = n, we know that a ring (Mn(F),+,⋅) is isomorphic to a ring (L(V ),+,∘)6. Therefore a ring Mn(F) is a regular ring. The following corollary follows from Corollary 4.
Let ℤ and ℕ denote the set of all integers and the set of all positive integers, respectively. For n ℕ, let (ℤn,+,⋅) 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 ℤn. The following proposition is well-known6.
Proposition 3 (Ehrlich8) For any n ℕ, the ring (ℤn,+,⋅) is regular if and only if n is square-free.
Then the following corollary is true.
Next, let n be a square-free number. By Proposition 3, the ring ℤn is regular.
1. Hickey JB (1983) Semigroups under a sandwich operation. Proc Edinb Math Soc 26, 371–82.
2. Magill KD Jr (1967) Semigroup structures for families of functions I. Some homomorphism theorems. J Aust Math Soc 7, 81–94.
3. Magill KD Jr (1967) Semigroup structures for families of functions II. Continuous functions. J Aust Math Soc 7, 95–107.
4. Hickey JB (1986) On variants of a semigroup. Bull Aust Math Soc 34, 199–212.
6. Hugerford TW (1974) Algebra, Springer-Verlag, New York.
8. Ehrlich G (1968) Unit-regular rings. Portugal Math 27, 209–12.