Remarks on finite pseudo-chain rings

We consider a class of local rings that is properly larger than that of chain rings and investigate its properties. We show that when restricted to finite rings, elements of such rings can be factorized into a finite product of irreducible elements, and the length of such factorization is unique, although the factorization itself is far from being unique. Using these results, we determine the minimal number of generators required for each ideal. We also show that several nontrivial examples of such rings appear as a subring of a chain ring and show that such rings can be constructed using techniques commonly used in the field of multiplicative ideal theory. We choose a class of such rings and investigate the basic properties of rings induced from them (including the number of elements and ideals and the unit group structure of such a ring), which are directly associated with the structure of cyclic codes over such rings.