Let $R$ be a ring. We recall that $R$ is called a near pseudo-valuation ring if every minimal prime ideal of $R$ is strongly prime. Let now $\sigma$ be an automorphism of $R$ and $\delta$ a $\sigma$-derivation of $R$. Then $R$ is said to be an almost $\delta$-divided ring if every minimal prime ideal of $R$ is $\delta$-divided. Let $R$ be a Noetherian ring which is also an algebra over $\mathbb{Q}$ ($\mathbb{Q}$ is the field of rational numbers). Let $\sigma$ be an automorphism of $R$ such that $R$ is a $\sigma (*)$-ring and $\delta$ a $\sigma$-derivation of $R$ such that $\sigma \left(\delta \right(a\left)\right)=\delta \left(\sigma \right(a\left)\right)$ for all $a\in R$. Further, if for any...

We study the class of modules which are invariant under idempotents of their envelopes. We say that a module M is -idempotent-invariant if there exists an -envelope u : M → X such that for any idempotent g ∈ End(X) there exists an endomorphism f : M → M such that uf = gu. The properties of this class of modules are discussed. We prove that M is -idempotent-invariant if and only if for every decomposition $X={⨁}_{i\in I}{X}_i$, we have $M={⨁}_{i\in I}\left(u^{-1}\left(X_i\right)\cap M\right)$. Moreover, some generalizations of -idempotent-invariant modules are considered....

We prove new necessary and sufficient conditions for a morphism of coalgebras to be a monomorphism, different from the ones already available in the literature. More precisely, φ: C → D is a monomorphism of coalgebras if and only if the first cohomology groups of the coalgebras C and D coincide if and only if ${\sum }_{i\in I}\epsilon \left(a^i\right)b^i={\sum }_{i\in I}{a}^i\epsilon \left(b^i\right)$ for all ${\sum }_{i\in I}{a}^i\otimes b^i\in C{◻}_{D}C$. In particular, necessary and sufficient conditions for a Hopf algebra map to be a monomorphism are given.

The so-called “invariance under twisting” for twisted tensor products of algebras is a result stating that, if we start with a twisted tensor product, under certain circumstances we can “deform” the twisting map and we obtain a new twisted tensor product, isomorphic to the given one. It was proved before that a number of independent and previously unrelated results from Hopf algebra theory are particular cases of this theorem. In this article we show that some more results from literature are particular...

Let $𝒳$ be a Banach space of dimension $n>1$ and $𝔄\subset ℬ\left(𝒳\right)$ be a standard operator algebra. In the present paper it is shown that if a mapping $d:𝔄\to 𝔄$ (not necessarily linear) satisfies $$d\left(\right[[U,V],W\left]\right)=\left[\right[d\left(U\right),V],W]+\left[\right[U,d\left(V\right)],W]+\left[\right[U,V],d(W\left)\right]$$ for all $U,V,W\in 𝔄$, then $d=\psi +\tau$, where $\psi$ is an additive derivation of $𝔄$ and $\tau :𝔄\to 𝔽I$ vanishes at second commutator $\left[\right[U,V],W]$ for all $U,V,W\in 𝔄$. Moreover, if $d$ is linear and satisfies the above relation, then there exists an operator $S\in 𝔄$ and a linear mapping $\tau$ from $𝔄$ into $𝔽I$ satisfying $\tau \left(\right[[U,V],W\left]\right)=0$ for all $U,V,W\in 𝔄$, such that $d\left(U\right)=SU-US+\tau \left(U\right)$ for all $U\in 𝔄$.

Let $R$ be a graded ring and $n\ge 1$ be an integer. We introduce and study the notions of Gorenstein $n$-FP-gr-injective and Gorenstein $n$-gr-flat modules by using the notion of special finitely presented graded modules. On $n$-gr-coherent rings, we investigate the relationships between Gorenstein $n$-FP-gr-injective and Gorenstein $n$-gr-flat modules. Among other results, we prove that any graded module in $R$-gr (or gr-$R$) admits a Gorenstein $n$-FP-gr-injective (or Gorenstein $n$-gr-flat) cover and preenvelope, respectively....