We introduce the notion of an automorphism liftable module and give a characterization to it. We prove that category equivalence preserves automorphism liftable. Furthermore, we characterize semisimple rings, perfect rings, hereditary rings and quasi-Frobenius rings by properties of automorphism liftable modules. Also, we study automorphism liftable modules with summand sum property (SSP) and summand intersection property (SIP).

A completely primary ring is a ring R with identity 1 ≠ 0 whose subset of zero-divisors forms the unique maximal ideal . We determine the structure of the group of automorphisms Aut(R) of a completely primary finite ring R of characteristic p, such that if is the Jacobson radical of R, then ³ = (0), ² ≠ (0), the annihilator of coincides with ² and $R/\cong GF\left(p^r\right)$, the finite field of $p^r$ elements, for any prime p and any positive integer r.

In this paper we improve recent results dealing with cellular covers of R-modules. Cellular covers (sometimes called colocalizations) come up in the context of homotopical localization of topological spaces. They are related to idempotent cotriples, idempotent comonads or coreflectors in category theory. Recall that a homomorphism of R-modules π: G → H is called a cellular cover over H if π induces an isomorphism $\pi ⁎:Ho{m}_R(G,G)\cong Ho{m}_R(G,H)$, where π⁎(φ) = πφ for each $\phi \in Ho{m}_R(G,G)$ (where maps are acting on the left). On the one hand,...

The main result: Let $R$ be a $2$-torsion free semiprime ring and let $T:R\to R$ be an additive mapping. Suppose that $T\left(xyx\right)=xT\left(y\right)x$ holds for all $x,y\in R$. In this case $T$ is a centralizer.

Soit $A_1\left(ℂ\right)$ la première algèbre de Weyl sur $ℂ$. La codimension B-W d’un idéal à droite non nul $I$ de $A_1\left(ℂ\right)$ a été introduite par Yuri Berest et George Wilson. Nous montrons d’une part que cette codimension est invariante par la relation de Stafford : si $x\in Q_1=\mathrm{Frac}\left(A_1\left(ℂ\right)\right)$, le corps de fractions de $A_1\left(ℂ\right)$, et si $\sigma \in \mathrm{Aut}\left(A_1\left(ℂ\right)\right)$, le groupe des $ℂ$-automorphismes de $A_1\left(ℂ\right)$, sont tels que $J=x\sigma \left(I\right)$ soit un idéal à droite de $A_1\left(ℂ\right)$, alors $\mathrm{codim}I=\mathrm{codim}x\sigma \left(I\right)$. Nous relions d’autre part la codimension d’un idéal $I$ à la codimension de Gail Letzter-Makar Limanov, de $\mathrm{End}\left(I\right)$, l’anneau des endomorphismes...

We develop a new combinatorial method to deal with a degree estimate for subalgebras generated by two elements in different environments. We obtain a lower bound for the degree of the elements in two-generated subalgebras of a free associative algebra over a field of zero characteristic. We also reproduce a somewhat refined degree estimate of Shestakov and Umirbaev for the polynomial algebra, which plays an essential role in the recent celebrated solution of the Nagata conjecture and the strong...