Starting from the notion of semistar operation, introduced in 1994 by Okabe and Matsuda [49], which generalizes classical concept star operation (cf. Gilmer's book [27]) and, hence, related theory ideal systems based on works W. Krull, E. Noether, H. Prüfer, P. Lorenzen Jaffard Halter–Koch's [32]), this paper we outline a general approach to Prüfer ⋆-multiplication domains (or P⋆MDs), where ⋆ is operation. This leads relax restriction base domain, not necessarily integrally closed case,...

We introduce and study the definition, main properties applications of iterated twisted tensor products algebras, motivated by problem defining a suitable representative for product spaces in noncommutative geometry. find conditions constructing an three factors prove that they are enough building any number factors. As example geometrical aspects our construction, we show how to construct differential forms involutions on starting from corresponding structures give some examples algebras...

Let $H$ be a Hopf algebra and let $\mathcal{M}_s (H)$ the category of all left $H$-modules right $H$-comodules satisfying appropriate compatibility relations. An object in will called stable anti-Yetter–Drinfeld module (over $H$) or SAYD , for short. To each $M \in \mathcal{M}_s we associate, functorial way, cyclic $\mathrm{Z}_\ast (H, M)$. We show that our construction can used to compute homology underlying structure relative $H$-Galois extensions. $K$ subalgebra $H$. For an arbitrary (K)$...

Let R be a semisimple ring. A pair (A,C) is called almost-Koszul if connected graded -ring and C compatible -coring. To an one associates three chain complexes cochain such that of them exact only the others are so. In this situation said to Koszul. One proves Koszul there -coring This result allows us investigate Hochschild (co)homology rings. We apply our method show twisted tensor product two rings More examples applications pairs, including generalization Fröberg Theorem [12], discussed...

Let H be a Hopf algebra. By definition modular crossed H-module is vector space M on which acts and coacts in compatible way. To every we associate cyclic object Z(H,M). The homology of Z(H,M) extends the usual algebra structure H, relative an H-Galois extension. For subalgebra K compute, under some assumptions, induced module. As direct application this computation, describe strongly graded algebras. In particular, calculate (usual) group algebras quantum tori. Finally, when enveloping Lie...

We study localizing and colocalizing subcategories of a comodule category coalgebra C over field, using the correspondence between equivalence classes idempotent elements in dual algebra C∗. In this framework, we give useful description localization functor by means Morita–Takeuchi context defined quasi-finite injective cogenerator subcategory. Applying description; first characterize that subcategory , with associated element e ∊ C∗, is if only eC eCe-comodule and, addition,  perfect...

The notion of the path coalgebra a quiver with relations introduced in \cite{simson1} and \cite{simson2} is studied. In particular, developing this topic context weak$^*$ topology, we give criterion that allows us to verify whether or n

For any commutative ring $A$ we introduce a generalization of $S$--noetherian rings using here\-ditary torsion theory $\sigma$ instead multiplicatively closed subset $S\subseteq{A}$. It is proved that totally noetherian w.r.t. local property, and if w.r.t $\sigma$, then finite type.

We study star-operations, in the sense of [7], on a domain and relate them to hereditary torsion theories. In first part we produce theories from after that some chain conditions relative star-operations. apply this kind multiplication domains: PVMD. final paper consider general ∗-multiplication domains, showing Prüfer may be characterized using class all τ-closed over-rings.

In a recent paper G. A. Cannon and K. M. Neuerburg point out that if $A=\mathbb{Z}$ $B$ is an arbitrary ring with unity, then $\mathbb{Z}\star{B}$, the Dorroh extension of $B$, isomorphic to direct product $\mathbb{Z}\times{B}$. Thus, ideal structure $\mathbb{Z}\star{B}$ can be completely described. The aim this note result may extended any pair $(A,B)$ in which $A$-algebra study construction extensions algebras without zero divisors their behavior respect algebra maps.

In this note we propose an effective method based on the computation of a Gröbner basis left ideal to calculate Gelfand-Kirillov dimension modules.

Abstract We study the “local” behavior of several relevant properties concerning semistar operations, like finite type, stable, spectral, e.a.b. and a.b. deal with “global” problem building a new operation on given integral domain, by “gluing” homogeneous family operations defined set localizations. apply these results for studying local–global Nagata ring Kronecker function ring. prove that an domain D is Prüfer ⋆-multiplication if only all its localizations P are ⋆ -multiplication domains.

For any commutative ring A, we introduce a generalization of S-artinian rings using hereditary torsion theory σ instead multiplicative closed subset S⊆A. It is proved that if A totally σ-artinian ring, then must be finite type, and σ-noetherian.