We find the universal module w(M) for functions (that need not be invariants) of finite matroids, defined on a minor-closed class M and with values in any module L over any commutative and unitary ring, that satisfy a parametrized deletion-contraction identity, F (M) = δeF (M r e) + γeF (M/e), when e is neither a loop nor a coloop. (F is called a (parametrized) weak Tutte function.) Within the ...

Abstract This paper is a sequel to T-structures and twisted complexes on derived injectives by the same author with W. Lowen M. Van den Bergh. We define dg-category of unbounded dg-category, which particularly interesting in case dg-categories or projectives associated t-structure. On such we natural “injective” dually “projective” intended as direct generalization homotopy categories injective...

Let R be a ring, a right ideal I of R is called small if for every proper right ideal K of R, I +K = R. A ring R is called right small injective if every homomorphism from a small right ideal to R R can be extended to an R-homomorphism from R R to R R. Properties of small injective rings are explored and several new characterizations are given for QF rings and P F rings, respectively.

let r be a commutative noetherian ring. we study the behavior of injectiveand at dimension of r-modules under the functors homr(-,-) and -×r-.

We consider the monoid Inj(M) of injective self-maps of a set M and want to determine its normal subsemigroups by numerical invariants. This was established by Mesyan in 2012 if M is countable. Here we obtain an explicit description of all normal subsemigroups of Inj(M) for any uncountable set M .

1. [10 points] Determine whether the following statements are true or false (you have to include proofs/counterexamples): (a) Let R be an integral domain, F – a free R-module of finite rank, and M – a torsion R-module. Then there is no injective homomorphism from F to M . Solution: True. Suppose there was an injective homomorpism φ : F → M . Then let N = φ(F ); N is a submodule of M , and there...

We present applications of contramodule techniques to the Enochs conjecture about covers and direct limits, both in categorical tilting context beyond. In n-tilting–cotilting correspondence situation, if $$\mathsf A$$ is a Grothendieck abelian category related B$$ equivalent contramodules over topological ring $$\mathfrak R$$ belonging one certain four classes rings (e. g., commutative), then l...

We construct a canonical pseudofunctor ( ) on the category of finite-type maps of (say) connected noetherian universally catenary finite-dimensional separated schemes, taking values in the category of Cousin complexes. This pseudofunctor is a concrete approximation to the restriction of the Grothendieck Duality pseudofunctor( ) to the full subcategory of the derived category having Cohen-Macaul...

We prove that if S is a properly embedded π1-injective surface in a compact 3-manifold M , then π1S is separable in π1M .