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Below, all rings are associative and unital; and the word "ideal" always refers to a two-sided ideal. Let's stipulate that a ring $R$ has property (P) if every non-unit of $R$ is contained in a comple …

Recall that a (unital) ring $R$ is von Neumann regular (VNR) if, for each $x \in R$, there exists $y \in R$ such that $x = xyx$; and unit-regular if such an element $y$ can be taken to be a unit. Que …

Let $R$ be a ring (throughout, all rings are associative and unital). We say $R$ satisfies condition (C) if, for every $a \in R$, there exists an integer $n \ge 1$ (depending on $a$) such that $a^n$ l …

Let $R$ be a ring (here, rings are always associative, unital, and non-zero). We say that $R$ is a left duo ring if $aR \subseteq Ra$ for every $a \in R$. Question. Is there a non-commutative, left d …

Let $T_n(R)$ be the ring of upper triangular $n$-by-$n$ matrices with entries in a (commutative or non-commutative) unital ring $R$. It happened to me to note that, if $R$ is a skew field, then $T_2(R …

[I fear that I'm missing something obvious here, but I'll dare to ask anyway.] As we all know, a division ring is a (unital, associative, non-zero) ring where every non-zero element is a unit. So, let …

Sorry for answering my own question, but it turned out that what I'm calling "anti-division rings" in the OP were already studied by P.M. Cohn under the name of "$0$-rings" (though Cohn's work on this …

Let $R$ be a (commutative or non-commutative) ring with identity. As usual, an idempotent $e \in R$ is primitive if $eR$ (the principal right ideal generated by $e$) is indecomposable as a right modul …

Let's say that a (right) module $M$ is well complemented if every non-zero submodule of $M$ has an indecomposable direct summand (by the way, is there a better or more standard name for this property? …

Let $R$ be a (commutative or non-commutative, associative) unital ring. Following Nicholson & Yousif [1, p. 21], we say that a right ideal $\mathfrak i$ of $R$ is extensive if every $R$-linear functio …

Let $R$ be a (commutative or non-commutative, associative) ring with unity, and let $a$ be an element of $R$ such that $r(a) = r(a^2)$, where $r(\cdot)$ denotes a right annihilator. It follows that $r …

[Sorry for answering my own question, and the more so because this is happening for the second time in 24 hours.] The question might be open. In fact, a positive answer would imply an equally positive …

Let $H$ be a (commutative or non-commutative) monoid. We say that $H$ satisfies the ACCPL (ascending chain condition on principal left ideals) if there exists no infinite sequence of principal left id …

A "close analogue" of (what I'm referring to as) Cohn's theorem on atomic factorizations in cancellative monoids (that is, Theorem 1 in the OP) is given by the unnumbered corollary on the bottom of p. …

A (unital) ring $R$ with the property that every element other than the identity $1_R$ is a (two-sided) zero divisor, seems to be commonly called a "$0$-ring" or "$\mathcal O$-ring". These rings were …