Decomposing elements of a right self-injective ring

TL;DR

This paper characterizes right self-injective rings where each element can be combined with a unit to produce units, extending previous results about sums of invertible elements in such rings.

Contribution

It establishes a new characterization of right self-injective rings based on the existence of units related to each element, specifically involving the rings' factor ring structures.

Findings

If R is right self-injective, then for each a in R, there exists a unit u with a+u and a−u units iff R has no factor ring isomorphic to Z_2 or Z_3.

The result generalizes earlier work on sums of invertible elements in rings.

The paper provides necessary and sufficient conditions for the existence of such units in right self-injective rings.

Abstract

It was proved independently by both Wolfson [An ideal theoretic characterization of the ring of all linear transformations, Amer. J. Math. 75 (1953), 358-386] and Zelinsky [Every Linear Transformation is Sum of Nonsingular Ones, Proc. Amer. Math. Soc. 5 (1954), 627-630] that every linear transformation of a vector space $V$ over a division ring $D$ is the sum of two invertible linear transformations except when $V$ is one-dimensional over $\mathbb Z_2$. This was extended by Khurana and Srivastava [Right self-injective rings in which each element is sum of two units, J. Algebra and its Appl., Vol. 6, No. 2 (2007), 281-286] who proved that every element of a right self-injective ring $R$ is the sum of two units if and only if $R$ has no factor ring isomorphic to $\mathbb Z_2$. In this paper we prove that if $R$ is a right self-injective ring, then for each element $a\in R$ there exists a unit $u\in R$ such that both $a+u$ and $a-u$ are units if and only if $R$ has no factor ring isomorphic to $\mathbb Z_2$ or $\mathbb Z_3$.