Elements in exchange rings with related comparability

Elements in Exchange Rings with Related Comparability

We show that if R is an exchange ring, then the following are equivalent: (1) R satisfies related comparability. (2) Given a,b,d ∈ R with aR+bR = dR, there exists a related unit w ∈ R such that a+bt = dw. (3) Given a,b ∈ R with aR = bR, there exists a related unit w ∈ R such that a= bw. Moreover, we investigate the dual problems for rings which are quasi-injective as right modules.

Two-Sided Properties of Elements in Exchange Rings

For any element a in an exchange ring R, we show that there is an idempotent e ∈ aR ∩ Ra such that 1 − e ∈ (1 − a)R ∩ R (1 − a). A closely related result is that a ring R is an exchange ring if and only if, for every a ∈ R, there exists an idempotent e ∈ Ra such that 1− e ∈ (1−a)R. The Main Theorem of this paper is a general two-sided statement on exchange elements in arbitrary rings which subs...

Nilpotent Elements in Skew Polynomial Rings

 Letbe a ring with an endomorphism and an -derivationAntoine studied the structure of the set of nilpotent elements in Armendariz rings and introduced nil-Armendariz rings. In this paper we introduce and investigate the notion of nil--compatible rings. The class of nil--compatible rings are extended through various ring extensions and many classes of nil--compatible rings are constructed. We al...

On constant products of elements in skew polynomial rings

Let $R$ be a reversible ring which is $alpha$-compatible for an endomorphism $alpha$ of $R$ and $f(X)=a_0+a_1X+cdots+a_nX^n$ be a nonzero skew polynomial in $R[X;alpha]$. It is proved that if there exists a nonzero skew polynomial $g(X)=b_0+b_1X+cdots+b_mX^m$ in $R[X;alpha]$ such that $g(X)f(X)=c$ is a constant in $R$, then $b_0a_0=c$ and there exist nonzero elements $a$ and $r$ in $R$ such tha...

Separative Ideals of Exchange Rings