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.
