Minimality ranks in the style of Deissler are one way of measuring the structural complexity of minimal extensions of first-order structures. In particular, positive Deissler rank measures the complexity of the injective envelope of a module as an extension of that module. In this paper we solve a problem of the second author by showing that certain injective envelopes have the maximum possible...

We find the injective hulls of partially ordered monoids in the category whose objects are po-monoids and submultiplicative order-preserving functions. These injective hulls are with respect to a special class of monics called “embeddings”. We show as well that the injective objects with respect to these embeddings are precisely the quantales.

In this paper, motivated by Cagliari and Mantovani, we have obtained a characterization of injective objects (with respect to the class embeddings in category

we introduce the class of “right almost v-rings” which is properly between the classes of right v-rings and right good rings. a ring r is called a right almost v-ring if every simple r-module is almost injective. it is proved that r is a right almost v-ring if and only if for every r-module m, any complement of every simple submodule of m is a direct summand. moreover, r is a right almost v-rin...

The axiom of choice ensures precisely that, in ZFC, every set is projective: that is, a projective object in the category of sets. In constructive ZF (CZF) the existence of enough projective sets has been discussed as an additional axiom taken from the interpretation of CZF in Martin-Löf’s intuitionistic type theory. On the other hand, every non–empty set is injective in classical ZF, which arg...

let r be a right gf-closed ring with finite left and right gorenstein global dimension. we prove that if i is an ideal of r such that r/i is a semi-simple ring, then the gorensntein flat dimensnion of r/i as a right r-module and the gorensntein injective dimensnnion of r/i as a left r-module are identical. in particular, we show that for a simple module s over a commutative gorensntein ring r, ...

It is known that injective Cellular Automata (CA) are surjective. In general, the converse is not true, and there are many algebraic CA counterexamples. However, there are interesting subclasses where this might be true. For example, if a surjective CA has entropy 0 then it is almost injective (Moothathu (2011)) and it is not known if it is actually injective. The author believes there are some...

Given two graphs G = (VG, EG) and H = (VH , EH), a homomorphism from G to H is a function f : VG → VH such that for every uv ∈ EG, f(u)f(v) ∈ EH . A homomorphism from G to H is referred to as an H-colouring of G and the vertices of H are regarded as colours. The graph H is called the target of the homomorphism. These definitions extend to directed graphs by requiring that the mapping must prese...