It is well known that every uniquely clean ring is strongly clean. In this paper, we investigate the question of when this result holds element-wise. We first construct an example showing that uniquely clean elements need not be strongly clean. However, in case every corner ring is clean the uniquely clean elements are strongly clean. Further, we classify the set of uniquely clean elements for ...

A ring R is called an E-ring if the canonical homomorphism from R to the endomorphism ring End (RZ) of the additive group RZ, taking any r ∈ R to the endomorphism left multiplication by r turns out to be an isomorphism of rings. In this case RZ is called an E-group. Obvious examples of E-rings are subrings of Q. However there is a proper class of examples constructed recently, see [8]. E-rings ...

A divisibility test of Arend Heyting, for polynomials over a …eld in an intuitionistic setting, may be thought of as a kind of division algorithm. We show that such a division algorithm holds for divisibility by polynomials of content 1 over any commutative ring in which nilpotent elements are zero. In addition, for an arbitary commutative ring R, we characterize those polynomials g such that t...

We identify some classes $$\mathcal {C}$$ of mixed groups such that if $$G\in \mathcal has the cancellation property then Walk-endomorphism ring G unit lifting property. In particular, is a self-small group torsion-free rank at most 4 with it decomposition $$G=F\oplus H$$ F free and H

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...

Let k be a field of characteristic p>0 and let H finite group or scheme. We show that the negative Tate cohomology ring Hˆ?0(H,k) can realized as endomorphism trivial module in Verdier localization stable category kG-modules for G an extension H. This means some cases is local with infinitely generated radical square zero. stands stark contrast to known calculations which degree zero component ...

We give a quasihomogeneity criterion for Gorenstein curves. For complete intersections, it is related to the first step of Vasconcelos’ normalization algorithm. In the process, we give a simplified proof of the Kunz–Ruppert criterion.