It it shown that the first-order theory of an automatic structure, whose Gaifman graph has bounded degree, is decidable in doubly exponential space (for injective automatic presentations, this holds even uniformly). Presenting an automatic structure of bounded degree whose theory is hard for 2EXPSPACE, we also prove this result to be optimal. These findings close the gap left open in [14].

in this paper‎, ‎we introduce the notion of $(m,n)$-‎algebr‎aically compact modules as an analogue of algebraically‎ ‎compact modules and then we show that $(m,n)$-algebraically‎ ‎compactness‎ ‎and $(m,n)$-pure injectivity for modules coincide‎. ‎moreover‎, ‎further characterizations of a‎ ‎$(m,n)$-pure injective module over a commutative ring are given‎.

Let $${\cal M}$$ be an n-cluster tilting subcategory of mod-Λ, where Λ is Artin algebra. S}({\cal M})$$ denote the full S}(\Lambda )$$ , submodule category Λ, consisting all monomorphisms in . We construct two functors from to $$\bmod - \underline {\cal M} $$ finitely presented additive contravariant on stable show that these are full, dense and objective hence provide equivalences between quot...

an r-module m is called strongly noncosingular if it has no nonzero rad-small (cosingular) homomorphic image in the sense of harada. it is proven that (1) an r-module m is strongly noncosingular if and only if m is coatomic and noncosingular; (2) a right perfect ring r is artinian hereditary serial if and only if the class of injective modules coincides with the class of (strongly) noncosingula...

We find a bound for the Goldie dimension of hereditary modules in terms of the cardinality of the generator sets of its quasi-injective hull. Several consequences are deduced. In particular, it is shown that every right hereditary module with countably generated quasi-injective hull is noetherian. Or that every right hereditary ring with finitely generated injective hull is artinian, thus answe...