Rings with minimum condition on principal ideals

Ohne Zusammenfassung     

Minimal presentations for monoids with the ascending chain condition on principal ideals

We show that the natural way to extend several key results concerning minimal presentations for finitely generated commutative cancellative reduced monoids, is to replace the finitely generated condition by the ascending chain condition on principal ideals.     

Orders in regular rings with minimal condition for principal right ideals

ABSTRACT

A ring R is called an n-clean (resp. Σ-clean) ring if every element in R is n-clean (resp. Σ-clean). Clean rings are 1-clean and hence are Σ-clean. An example shows that there exists a 2-clean ring that is not clean. This shows that Σ-clean rings are a proper generalization of clean rings. The group ring ?_(p) G with G a cyclic group of order 3 is proved to be Σ-clean. The m× m matrix ring M _m (R) over an n-clean ring is n-clean, and the m×m (m>1) matrix ring M _m (R) over any ring is Σ-clean. Additionally, rings satisfying a weakly unit 1-stable range were introduced. Rings satisfying weakly unit 1-stable range are left-right symmetric and are generalizations of abelian π-regular rings, abelian clean rings, and rings satisfying unit 1-stable range. A ring R satisfies a weakly unit 1-stable range if and only if whenever a ₁ R + ˙˙˙ a _m R = dR, with m ≥ 2, a ₁,…, a _m, d ∈ R, there exist u ₁ ∈ U(R) and u ₂,…, u _m ∈ W(R) such that a ₁ u ₁ + ? a _m u _m = Rd.     

Finite rings of principal ideals

We investigate a class of finite rings of principal ideals: we derive the necessary and sufficient conditions that the rings of this class are uniquely defined by certain parameters to within an isomorphism.Translated from Matematicheskie Zametki, Vol. 12, No. 4, pp. 373–379, October, 1972.In conclusion the author wishes to thank V. P. Elizarov for formulating the problem and for attention and help in the preparation of the paper.     

Rings with projective principal right ideals

It has been proved that if A is a right-distributive ring, algebraic over its center, and whose principal ideals are projective, then A is a left-distributive ring.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 6, pp. 861–863, June, 1990.     

Prime and principal ideals in the algebraN +

Archiv der Mathematik -     

On generic principal ideals in the exterior algebra

We give a lower bound on the Hilbert series of the exterior algebra modulo a principal ideal generated by a generic form of odd degree and disprove a conjecture by Moreno-Socías and Snellman. We also show that the lower bound is equal to the minimal Hilbert series in some specific cases.     

Powers of t -spread principal Borel ideals

We prove that t-spread principal Borel ideals are sequentially Cohen–Macaulay and study their powers. We show that these ideals possess the strong persistence property and compute their limit depth.     

An Engel condition with derivation for left ideals

We generalize a number of results in the literature by proving the following theorem: Let be a semiprime ring, a nonzero derivation of , a nonzero left ideal of , and let . If for some positive integers , and all , the identity holds, then either or else the ideal of generated by and is in the center of . In particular, when is a prime ring, is commutative.

     

A condition for diagonalization modulo arbitrary norm ideals

We give a spectral condition which is sufficient for the simultaneous diagonalization of a commuting tuple of self-adjoint operators modulo a given norm ideal. For diagonalization modulo certain norm ideals this condition is also necessary, while for other norm ideals this condition seems to be close to being necessary. Moreover, this condition is easy to verify in applications.     

On form ideals and Artin-Rees condition

We look for conditions which make two ideals and in a noetherian ring A have the same form ideal in an associated graded ring G_A(α). More precisely, when and and f_i?f'_iεα ^m m?0, ? i , we give a necessary and sufficient condition to have , involving the first syzygies modules both of (f₁,...,f_n) and (f'₁,...,f'_n); our proof is based on the Artin-Rees lemma. Finally we show that, when the sequence f₁,...,f_n is regular and for an integer q, then f₁?f'_i ? α^q+1 ? i implies .     

Notes on a new chain condition for prime ideals

A chain condition intermediate to the catenary property and the chain condition for prime ideals (c.c.) is studied. Like the c.c., the condition is inherited from a semi-local domain R by integral extension domains, by local quotient domains, and by factor domains, and a semi-local ring that satisfies the condition is catenary. (Unlike the c.c., none of these statements is true when R is not semi-local.) A number of characterizations of a semi-local domain that satisfies the condition are given in terms of: integral (respectively, algebraic, transcendental) extension domains, Henselizations, completions, Rees rings, associated graded rings and certain discrete valuation over-rings. Then four of the catenary chain conjectures are characterized in terms of this condition.     

An Engel condition with skew derivations for one-sided ideals

We apply the theory of generalized polynomial identities with automorphisms and skew derivations to prove the following theorem: Let A be a prime ring with the extended centroid C and with two-sided Martindale quotient ring Q, R a nonzero right ideal of A and \(\delta \) a nonzero \(\sigma \)-derivation of A, where \(\sigma \) is an epimorphism of A. For \(x,y\in A\), we set \([x,y] = xy - yx\). If \([[\ldots [[\delta (x^{n_0}),x^{n_1}],x^{n_{2}}],\ldots ],x^{n_k}]=0\) for all \(x\in R\), where \(n_{0},n_{1},\ldots ,n_{k}\) are fixed positive integers, then one of the following conditions holds: (1) A is commutative; (2) \(C\cong GF(2)\), the Galois field of two elements; (3) there exist \(b\in Q\) and \(\lambda \in C\) such that \(\delta (x)=\sigma (x)b-bx\) for all \(x\in A\), \((b-\lambda )R=0\) and \(\sigma (R)=0\). The analogous result for left ideals is also obtained. Our theorems are natural generalizations of the well-known results for derivations obtained by Lanski (Proc Am Math Soc 125:339–345, 1997) and Lee (Can Math Bull 38:445–449, 1995).     

Ideals that are an irredundant union of principal ideals

We investigate ideals of a commutative ring that are an irredundant union of principal ideals. Special attention is paid to prime ideals that are a finite union of principal ideals.