Let f : X → Y be a Cohen-Macaulay map of finite type between Noetherian schemes, and g : Y ′ → Y a base change map, with Y ′ Noetherian. Let f ′ : X → Y ′ be the base change of f under g and g : X → X the base change of g under f . We show that there is a canonical isomorphism θ g : g ∗ ωf ≃ ωf ′ , where ωf and ωf ′ are the relative dualizing sheaves. The map θ g f is easily described when f is...

Aghapournahr and Melkersson introduced the notion of condition on a Serre subcategory category modules over commutative Noetherian ring. This paper investigates structure set prime ideals satisfying subcategory. We try to calculate members this for consisting extension in two given subcategories. Moreover, we classify sets rings with small dimension.

Let $R$ be a commutative Noetherian ring with non-zero identity and $fa$ an ideal of $R$. Let $M$ be a finite $R$--module of finite projective dimension and $N$ an arbitrary finite $R$--module. We characterize the membership of the generalized local cohomology modules $lc^{i}_{fa}(M,N)$ in certain Serre subcategories of the category of modules from upper bounds. We define and study the properti...

Skew polynomial rings have invited attention of mathematicians and various properties of these rings have been discussed. The nature of ideals (in particular prime ideals, minimal prime ideals, associated prime ideals), primary decomposition and Krull dimension have been investigated in certain cases. In this article, we introduce a notion of primary decomposition of a noncommutative ring. We s...

Let G = GLN or SLN as reductive linear algebraic group over a field k of characteristic p > 0. We prove several results that were previously established only when N ≤ 5 or p > 2 : Let G act rationally on a finitely generated commutative k-algebra A and let grA be the Grosshans graded ring. We show that the cohomology algebra H(G, grA) is finitely generated over k. If moreover A has a good filtr...

Zaks (1969) proved that the answer is affirmative for a left and right noetherian ring if both dimensions are finite. Such rings are called Gorenstein. For a positive integer k, Auslander and Reiten (1994) initiated the study of k-Gorenstein algebras, which has stimulated several investigations. They showed that the answer to the question above is positive in case is an artin -Gorenstein algebr...

Let $R$ be a commutative ring with identity and let $M$ be an $R$-module. A proper submodule $P$ of $M$ is called strongly prime submodule if $(P + Rx : M)ysubseteq P$ for $x, yin M$, implies that $xin P$ or $yin P$. In this paper, we study more properties of strongly prime submodules. It is shown that a finitely generated $R$-module $M$ is Artinian if and only if $M$ is Noetherian and every st...

In this paper, an algebraic theory for local rings of finite embedding dimension is developed. Several extensions of (Krull) dimension are proposed, which are then used to generalize singularity notions from commutative algebra. Finally, variants of the homological theorems are shown to hold in equal characteristic. This theory is then applied to Noetherian local rings in order to get: (i) over...

Let T be a Noetherian ring and f a nonzerodivisor on T . We study concrete necessary and sufficient conditions for a module over R = T/(f) to be weakly liftable to T , in the sense of Auslander, Ding and Solberg. We focus on cyclic modules and get various positive and negative results on the lifting and weak lifting problems. For a module over T we define the loci for certain properties: liftab...

A condition on an affine central subalgebra Z of a noetherian algebra A of finite GelfandKirillov dimension, which we call here unruffledness, is shown to be equivalent in some circumstances to the flatness of A as a Z-module. Unruffledness was studied by Borho and Joseph in work on enveloping algebras of complex semisimple Lie algebras, and we discuss applications of our result to enveloping a...