On classes of rings related to the Veldsman classes

Abstract In general, the property of being an ideal-whether left, right, or two-sided-is not transitive in rings, a fact that has motivated significant research. In this paper, we introduce and investigate a new class of rings, denoted by $$\mathcal {K}_{1}$$ , defined by the condition that if J is a left ideal of a left ideal L of a ring R , and also a right ideal of a right ideal K of R , then J is a left ideal of R . We show that $$\mathcal {K}_{1}$$ is distinct from the Veldsman classes $$\mathcal {K} (x,y;z)$$ by Veldsman (Publ Math Debrecen 38(3–4):297–309, 1991) and we establish its position relative to them through strict inclusion relations. Several characterisations and properties of the rings in $$\mathcal {K}_{1}$$ are given. In particular, we present analogues of known results for the Veldsman classes: for example, we describe rings in $$\mathcal {K}_{1}$$ that are direct sums of copies of a ring, investigate their Baer radical and identify and study some properties of the largest subclass of $$\mathcal {K}_{1}$$ that is closed under direct sums.