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 K₁, 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 K₁ is distinct from the Veldsman classes 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 K₁ are given. In particular, we present analogues of known results for the Veldsman classes: for example, we describe rings in K₁ that are direct sums of copies of a ring, investigate their Baer radical and identify and study some properties of the largest subclass of K₁ that is closed under direct sums.
Deolinda Isabel Mendes (Mon,) studied this question.