the main objective of this study is to swing krull intersection theorem in primary decomposition of rings and modules to the primary decomposition of soft rings and soft modules. to fulfill this aim several notions like soft prime ideals, soft maximal ideals, soft primary ideals, and soft radical ideals are introduced for a soft ring over a given unitary commutative ring. consequently, the p...

Let M be a module over an associative ring R and σ[M ] the category of M -subgenerated modules. Generalizing the notion of a projective generator in σ[M ], a module P ∈ σ[M ] is called tilting in σ[M ] if (i) P is projective in the category of P -generated modules, (ii) every P -generated module is P presented, and (iii) σ[P ] = σ[M ]. We call P self-tilting if it is tilting in σ[P ]. Examples ...

Let G be a finite group and let k be a field of characteristic p. It is known that a kG-module V carries a non-degenerate G-invariant bilinear form b if and only if V is self-dual. We show that whenever a Morita bimodule M which induces an equivalence between two blocks B(kG) and B(kH) of group algebras kG and kH is self-dual then the correspondence preserves self-duality. Even more, if the bil...

Let R be a ring and R a self-orthogonal module. We introduce the notion of the right orthogonal dimension (relative to R ) of modules. We give a criterion for computing this relative right orthogonal dimension of modules. For a left coherent and semilocal ring R and a finitely presented self-orthogonal module R , we show that the projective dimension of R and the right orthogonal dimension (rel...

In this paper we propose a research programme for getting structural characterisations for 2-dimensional languages generated by self-assembling tiles. This is part of a larger programme on getting a formal foundation of parallel, interactive, distributed systems.