We introduce the "$type{lffs}$ subdirect product" and show that every ring is uniquely a $type{lffs}$ subdirect product of a family of $simple{basicls}$ rings. Also we show some applications.

We investigate the structure of subdirect products of groups, particularly their finiteness properties. We pay special attention to the subdirect products of free groups, surface groups and HNN extensions. We prove that a finitely presented subdirect product of free and surface groups virtually contains a term of the lower central series of the direct product or else fails to intersect one of t...

Bandelt and Petrich (1982) proved that an inversive semiring S is a subdirect product of a distributive lattice and a ring if and only if S satisfies certain conditions. The aim of this paper is to obtain a generalized version of this result. The main purpose of this paper however, is to investigate, what new necessary and sufficient conditions need we impose on an inversive semiring, so that, ...

Primeness on modules can be defined by prime elements in a suitable partially ordered groupoid. Using a product on the lattice of submodules L(M) of a module M defined in [3] we revise the concept of prime modules in this sense. Those modules M for which L(M) has no nilpotent elements have been studied by Jirasko and they coincide with Zelmanowitz’ “weakly compressible” modules. In particular w...

We study subdirect products of free and Demushkin pro-p groups of depth ∞ developing theory similar to the abstract case, see [4]. Furthermore we classify when a subdirect product has homological type FPm for some m ≥ 2, a problem still open for abstract groups for m ≥ 3.

It is shown that any n-chromatic graph is a full subdirect product of copies of the complete graphs Kn and Kn+1, except for some easily described graphs which are full subdirect products of copies of Kn+1 − {◦–◦} and Kn+2 − {◦–◦}; full means here that the projections of the decomposition are epimorphic in edges. This improves some results of Sabidussi. Subdirect powers of Kn or Kn+1 − {◦–◦} are...

In this paper we study conjugacy separability of subdirect products of two free (or hyperbolic) groups. We establish necessary and sufficient criteria and apply them to fibre products to produce a finitely presented group G1 in which all finite index subgroups are conjugacy separable, but which has an index 2 overgroup that is not conjugacy separable. Conversely, we construct a finitely present...