Definition 2.1. A commutative ring R is Noetherian if every chain of ideals in R I0 ⊂ I1 ⊂ I2 ⊂ · · · terminates after a finite number of steps (i.e., there is an interger k such that Is = Ik if s ≥ k). Remark 2.2. Polynomial rings R [X1, . . . , Xn] are Noetherian if R is. In particular, S (p) = C [p∗] is Noetherian. Theorem 2.3. If R is a Noetherian ring, and M is a finitely generated R-modul...

‎In this paper we give an upper bound for Noetherian dimension of all injective modules with Krull dimension on arbitrary rings‎. ‎In particular‎, ‎we also give an upper bound for Noetherian dimension of all Artinian modules on Noetherian duo rings.

We consider the Noetherian properties of the ring of differential operators of an affine semigroup algebra. First we show that it is always right Noetherian. Next we give a condition, based on the data of the difference between the semigroup and its scored closure, for the ring of differential operators being anti-isomorphic to another ring of differential operators. Using this, we prove that t...

In this paper we introduce a new class of modules which is closely related to the class of Noetherian modules. Let $R$ be a commutative ring with identity and let $M$ be an $R$-module such that $Nil(M)$ is a divided prime submodule of $M$. $M$ is called a Nonnil-Noetherian $R$-module if every nonnil submodule of $M$ is finitely generated. We prove that many of the properties of Noetherian modul...

One of our main results is a classification all the possible quivers of selfinjective radical cube zero finite dimensional algebras over an algebraically closed field having finite complexity. In the paper [5] we classified all weakly symmetric algebras with support varieties via Hochschild cohomology satisfying Dade’s Lemma. For a finite dimensional algebra to have such a theory of support var...

The long standing Lech’s conjecture in commutative algebra states that for a flat local extension $$(R,\mathfrak {m})\rightarrow (S,\mathfrak {n})$$ of Noetherian rings, we have an inequality on the Hilbert–Samuel multiplicities: $$e(R)\le e(S)$$ . In general is wide open when $$\dim R>3$$ , even equal characteristic. this paper, prove all dimensions, provided {m})$$ standard graded ring over p...

We determine the relationship between the cardinality of a Noetherian integral domain and the cardinality of a residue field. One consequence of the main result is that it is provable in ZFC that there is a Noetherian domain of cardinality א1 with a finite residue field, but the statement “There is a Noetherian domain of cardinality א2 with a finite residue field” is equivalent to the negation ...

We study the combination of the following already known ideas for showing connuence of unconditional or conditional term rewriting systems into practically more useful connuence criteria for conditional systems: Our syntactic separation into constructor and non-constructor symbols, Huet's introduction and Toyama's generalization of parallel closedness for non-noetherian unconditional systems, t...

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...

We initiate a unified, axiomatic study of noncommutative algebras R whose prime spectra are, in a natural way, finite unions of commutative noetherian spectra. Our results illustrate how these commutative spectra can be functorially “sewn together” to form SpecR. In particular, we construct a bimodule-determined functor ModZ → ModR, for a suitable commutative noetherian ring Z, from which there...