38. Following the hint, suppose I is a non-zero ideal (since the zero ideal, (0), is clearly principal). Then there exists some a ∈ I with a 6= 0. Then, either a is positive, or −a is positive, and also in I since I is a subring of Z. Thus, I contains positive elements. Now, let S = {x ∈ I |x > 0}. By the Well-Ordering Property of Z, S contains a smallest element, call it c. We claim that, in f...

Note that if ∑ agg ∈ I, then ∑ ag = 0 so that ∑ agg = ∑ agg− ∑ ag1 = ∑ ag(g−1), so that every element in I is a Z-linear combination of the elements g− 1 (g ∈ G, g 6= 1). Also, if ∑ ag(g− 1) = 0, it follows that ag = 0 for all g so that the elements g − 1 (g ∈ G, g 6= 1) are linearly independent. (b) Notice that the ideal generated by the elements s− 1 (s ∈ S) is the same as the ideal generated...

We consider extensions of unital commutative rings. We define an extension R ↪→ S to be a p-extension if every principally generated ideal of S is generated by an element of R. Examples are plentiful and localizations of regular multiplicative sets are p-extensions. We develop the theory of pextensions.

0. Introduction. The literature on algebraic semigroups contains publications concerning two distinct types of semigroup extensions. Ideal extensions of semigroups were introduced by Clifford in [1] whereas Rédei [9] introduced Schreier extensions of monoids (a monoid is a semigroup containing an identity element). Let A and 5 be two disjoint semigroups and let 5" contain a zero element o. A se...

The inherent bistability and picosecond time-scale switching of the resonant tunneling diode (RTD) provides an ideal element for the design of digital circuits and analog signal quantizers in the 10-100 GHz domain. New differential RTD-based circuits for quantizers and a first-order Sigma-Delta modulator capable of operating at 10 GHz and beyond are

We consider extensions of unital commutative rings. We define an extension R ↪→ S to be a p-extension if every principally generated ideal of S is generated by an element of R. Examples are plentiful and localizations of regular multiplicative sets are p-extensions. We develop the theory of pextensions.

In this article we prove that for a basic classical Lie superalgebra the annihilator of a strongly typical Verma module is a centrally generated ideal. For a basic classical Lie superalgebra of type I we prove that the localization of the enveloping algebra by a certain central element is free over its centre.

let $r$ be a commutative ring with identity and $mathbb{a}(r)$ be the set   of ideals of $r$ with non-zero annihilators. in this paper, we first introduce and investigate the principal ideal subgraph of the annihilating-ideal graph of $r$, denoted by $mathbb{ag}_p(r)$. it is a (undirected) graph with vertices $mathbb{a}_p(r)=mathbb{a}(r)cap mathbb{p}(r)setminus {(0)}$, where   $mathbb{p}(r)$ is...

Two classical results of Stafford say that every (left) ideal of the n-th Weyl algebra An can be generated by two elements, and every holonomic An-module is cyclic, i.e. generated by one element. We modify Stafford’s original proofs to make the algorithmic computation of these generators possible.

In this paper we study the relationships among the spectra of the cosets of an element of a Banach algebra in some quotient algebras. We also characterize the spectrum of any a E M (where M is an ideal of a Banach algebra with identity and moreover has an identity) in the whole algebra in terms of the spectrum of a in M.