Inspired by the concept of regular local rings in classical algebra, this article we initiate study parameter elements a commutative Noetherian hyperring. These provide deep connection between dimension hyperring and its primary hyperideals. Then, our focusses on R, with maximal hyperideal M, having property that R is equal to vectorial hyperspace MM2 over hyperfield RM. Finally, using hyperrin...

Abstract It is proven that finite idempotent left non-degenerate set-theoretic solutions $(X,r)$ of the Yang–Baxter equation on a set $X$ are determined by simple semigroup structure (in particular, union isomorphic copies group) and some maps $q$ $\varphi _{x}$ $X$, for $x\in X$. This turns out to be group precisely when associated monoid $M(X,r)$ cancellative all equal an automorphism this gr...

The Noetherian type of a space is the least κ for which the space has a κop-like base, i.e., a base in which no element has κ-many supersets. We prove some results about Noetherian types of (generalized) ordered spaces and products thereof. For example: the density of a product of not-too-many compact linear orders never exceeds its Noetherian type, with equality possible only for singular Noet...

For a coalgebraC , the rational functor Rat(−) : C∗ → C∗ is a left exact preradical whose associated linear topology is the family C , consisting of all closed and cofinite right ideals of C∗. It was proved by Radford (1973) that if C is right Noetherian (which means that every I ∈ C is finitely generated), then Rat(−) is a radical. We show that the converse follows if C1, the second term of th...

It is proved that if $\varphi\colon A\to B$ a local homomorphism of commutative noetherian rings, nonzero finitely generated $B$-module $N$ whose flat dimension over $A$ at most $\mathrm{edim}\, A - \mathrm{edim}\, B$, free $B$, and $\varphi$ special type complete intersection. This result motivated by "patching method" developed Taylor Wiles, conjecture de Smit, the first author, dealing with ...

A result of Artin, Small, and Zhang is used to show that a noetherian algebra over a commutative, noetherian Jacobson ring will be Jacobson if the algebra possesses a locally finite, noetherian associated graded ring. This result is extended to show that if an algebra over a commutative noetherian ring has a locally finite, noetherian associated graded ring, then the intersection of the powers ...

In this paper we define and explore the analytic spread ℓ ( I ) $\ell (\mathcal {I})$ of a filtration in local ring. We show that, especially for divisorial symbolic filtrations, some basic properties an ideal extend to even when is non-Noetherian. also illustrate significant differences between with examples. case $I$ , have classical bounds ht ⩽ dim R $\mbox{ht}(I)\leqslant \ell (I)\leqslant ...

Let k be an uncountable algebraically closed field and let A be a countably generated left Noetherian k-algebra. Then we show that A⊗k K is left Noetherian for any field extension K of k. We conclude that all subfields of the quotient division algebra of a countably generated left Noetherian domain over k are finitely generated extensions of k. We give examples which show that A⊗k K need not re...

We investigate two invariants of Noetherian semiperfect rings, namely the depth and a new invariant we call “delooping level”. These give lower upper bounds for finitistic dimension, respectively. As first theorems, necessary sufficient criterion delooping level to be finite in terms splitting unit map related adjunction, use this torsionfreeness finiteness. further relate these Auslander–Bridg...