A remark on almost complete intersections

Let k be a perfect field and S the quotient ring of a polynomial ring k[X₁,...,X_t] with respect to a prime ideal. Let I be a prime ideal of S such that R=S/I is an almost complete intersection. Then, in his paper [2], Matsuoka proves that the homological dimension of the differential module _R/Kis infinite under the assumption that R is Cohen-Macaulay and I² is a primary idea]. In this paper we prove that the result is valid without the above assumption.     

über Homomorphismen Projektiver Hjelmslev-Ebenen

The main result of this paper is a representation theorem for incidence morphisms of desarguesian Hjelmslev planes which preserve basis quadrangles. We prove that each geometric morphism between desarguesian Hjelmslev planes (R), (S) induces a total order of the coordinate ring R and a partial homomorphism from R to S. Conversely we have for each partial homomorphism and every partial order a uniquely determined geometric morphism. By a total order A of a ring R we mean a subring of R such that the elements of R/A are units in R, with inverses lying in A. If we have such a total order A \( \subseteq \) R, a partial homomorphism from R to another ring S is essentially a homomorphism from A to S.     

Quotient rings of polynomial rings

Let R be a commutative ring with identity. The multiplicatively closed sets U₂={fR[X]: c(f)–1=R}, (U₂)={fU₂: f is regular} and S={fR[X]: c(f)=R} are studied. By considering various equalities between these sets, many characterizations of Noetherian rings are found. In particular, a Noetherian ring R has depth 1 if and only if S=(U₂): and each maximal ideal of a Noetherian ring is regular if and only if U₂=(U₂).The theory of Prüfer v-multiplication rings (PVMR's) is developed for rings with zero divisors. Six equivalent conditions are given to the statement that an additively regular v-ring R is a PVMR.     

Normalizer of a group of “diagonal” automorphisms in algebras over a commutative ring

Let S be a faithful algebra over commutative ring R. It is assumed that S is additively generated by its invertible elements. It is shown that the nomalizer of subgroup Aut(S_s) of group Aut(S_R) coincides with the semidirect product Aut(S_S) Aut(S/R),where the second factor is the group of all ring automorphisms of ring S identical on R.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademiya Nauk SSSR, Vol. 191, pp. 5–8, 1991.     

Classifying Serre subcategories of finitely presented modules

Given a commutative coherent ring , a bijective correspondence between the thick subcategories of perfect complexes and the Serre subcategories of finitely presented modules is established. To construct this correspondence, properties of the Ziegler and Zariski topologies on the set of isomorphism classes of indecomposable injective modules are used in an essential way.

     

On Directly Infinite Rings

A ring R with identity 1 is said to be directly finite if for any a, b R, ab = 1 implies that ba = 1; otherwise R is directly infinite. With N the set of nonnegative integers, let B be the ring of N x N matrices over the ring of integers generated by two particular matrices. Properties of directly infinite rings are explored in relation to the ring B. This is made possible by various characterizations of the ring B one of which is that it is torsion free and generated by a bicyclic subsemigroup of its multiplicative semigroup. Some ideals and all idempotents of the ring B are constructed. The concepts of directly finite and chain finite idempotents are introduced in an arbitrary ring and applied to the ring B.     

Fast convolutions meet Montgomery

Arithmetic in large ring and field extensions is an important problem of symbolic computation, and it consists essentially of the combination of one multiplication and one division in the underlying ring. Methods are known for replacing one division by two short multiplications in the underlying ring, which can be performed essentially by using convolutions.

However, while using school-book multiplication, modular multiplication may be grouped into operations (where denotes the number of operations of one multiplication in the underlying ring), the short multiplication problem is an important obstruction to convolution. It raises the costs in that case to . In this paper we give a method for understanding and bypassing this problem, thus reducing the costs of ring arithmetic to roughly when also using fast convolutions. The algorithms have been implemented with results which fit well the theoretical prediction and which shall be presented in a separate paper.

     

Picard groups and infinite matrix rings

We describe a connection between the Picard group of a ring with local units and the Picard group of the unital overring . Using this connection, we show that the three groups , , and are isomorphic for any unital ring . Furthermore, each element of arises from an automorphism of , which yields an isomorphsm between and . As one application we extend a classical result of Rosenberg and Zelinsky by showing that the group is abelian for any commutative unital ring .

     

Matlis duals of top Local cohomology modules

In the first section of this paper we present generalizations of known results on the set of associated primes of Matlis duals of local cohomology modules; we prove these generalizations by using a new technique. In section 2 we compute the set of associated primes of the Matlis dual of , where is a -dimensional local ring and an ideal such that and .

     

On almost complete intersections

Let R be a local ring such that R=S/I where S is a regular local ring and I is a prime ideal of height r. In this paper it is shown that if I is minimally generated by r+1 elements, then there exists an R-homomorphism : K_RR^r+1 such that is an injection and R^r+1/(K_R)I/I² where K_R:=Ext _S ^r (R,S) the canonical module of R. Moreover, in case where S is a locality over a perfect field k, it is also shown that if R is Cohen-Macaulay and I² is a primary ideal, then the homological dimension of the differential module _R/k is infinite.The author wishes to thank his colleague Mr.Y.Aoyama for valuable discussions in connection with this subject.     

Universally meager sets

We study category counterparts of the notion of a universal measure zero set of reals.

We say that a set is universally meager if every Borel isomorphic image of is meager in . We give various equivalent definitions emphasizing analogies with the universally null sets of reals.

In particular, two problems emerging from an earlier work of Grzegorek are solved.

     

The dynamical properties of Penrose tilings

The set of Penrose tilings, when provided with a natural compact metric topology, becomes a strictly ergodic dynamical system under the action of by translation. We show that this action is an almost 1:1 extension of a minimal action by rotations on , i.e., it is an generalization of a Sturmian dynamical system. We also show that the inflation mapping is an almost 1:1 extension of a hyperbolic automorphism on . The local topological structure of the set of Penrose tilings is described, and some generalizations are discussed.

     

Induktive Limiten endlich erzeugter freier Moduln

If X is a set, R a ring and M an R-module, then it is shown that the R-module of all M-valued functions on X taking on X only finitely many values is free over R. This result is applied to show that inductive limits of finitely generated free R-modules are free over R, provided that the mappings of the inductive system are of a certain diagonal type. No restrictions are placed on the index set. There is a dual statement for projective limits. Among other applications it is shown that the -1^st Borel-Moore homology group (with values in any sheaf over an arbitrary integral domain and taken with respect to any paracompactifying family of supports) of a locally compact space vanishes.     

The maximum condition on annihilators for polynomial rings

For each positive integer , we construct a commutative ring such that the polynomial ring satisfies the maximum condition on annihilators and does not. In particular, there exists a commutative Kerr ring such that is not Kerr. This answers in the negative a question of Faith's.

     

The heart of prime ideals in Ore extensions

This article deals with prime ideals in Ore extensions S=R_α[X] over the right Noetherian K-algebra R. More specifically, we try to determine the structure of the centre of the classical ring of quotients of S modulo a prime ideal P from corresponding data in R. The main result asserts that is always a finitely generated field extension of K, provided this holds for the prime ideals in R. This yields a somewhat more elementary proof of a result in [6] which states that for any prime P in the group algebra R=K[G] of a polycyclic-by-finite group G, is a finitely generated field extension of K.     

Noncommutative Valuation Rings of the Quotient Artinian Ring of a Skew Polynomial Ring

Let R be a Dubrovin valuation ring of a simple Artinian ring Q and let Q[X,] be the skew polynomial ring over Q in an indeterminate X, where is an automorphism of Q. Consider the natural map from Q[X,]_XQ[X,] to Q, where Q[X,]_XQ[X,] is the localization of Q[X,] at the maximal ideal XQ[X,] and set , the complete inverse image of R by . It is shown that is a Dubrovin valuation ring of Q(X,) (the quotient ring of Q[X,]) and it is characterized in terms of X and Q. In the case where R is an invariant valuation ring, the given automorphism is classified into five types, in order to study the structure of (the value group of ). It is shown that there is a commutative valuation ring R with automorphism which belongs to each type and which makes Abelian or non-Abelian. Furthermore, some examples are used to show that several ideal-theoretic properties of a Dubrovin valuation ring of Q with finite dimension over its center, do not necessarily hold in the case where Q is infinite-dimensional. Presented by A. VerschorenMathematics Subject Classifications (2000) 16L99, 16S36, 16W60.     

Rational operators of the space of formal series

The main result of this paper is the following theorem: the group ring of the universal covering of the group SL(2, ℝ) is embeddable in a skew field with valuation in the sense of Mathiak and the valuation ring is an exceptional chain order in the skew field , i.e., there exists a prime ideal that is not completely prime. In this ring, every divisorial right fractional ideal is principal, and the linearly ordered set of all divisorial fractional right ideals is isomorphic to the real line. This theorem is a consequence of the fact that the universal covering group satisfies sufficient conditions for the embeddability of the group ring of a left ordered group in a skew field. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 3, pp. 9–53, 2006.     

Geometric cohomology frames on Hausmann-Holm-Puppe conjugation spaces

For certain manifolds with an involution the mod 2 cohomology ring of the set of fixed points is isomorphic to the cohomology ring of the manifold, up to dividing the degrees by two. Examples include complex projective spaces and Grassmannians with the standard antiholomorphic involution (with real projective spaces and Grassmannians as fixed point sets).

Hausmann, Holm and Puppe have put this observation in the framework of equivariant cohomology, and come up with the concept of conjugation spaces, where the ring homomorphisms arise naturally from the existence of what they call cohomology frames. Much earlier, Borel and Haefliger had studied the degree-halving isomorphism between the cohomology rings of complex and real projective spaces and Grassmannians using the theory of complex and real analytic cycles and cycle maps into cohomology.

The main result in the present note gives a (purely topological) connection between these two results and provides a geometric intuition into the concept of a cohomology frame. In particular, we see that if every cohomology class on a manifold with involution is the Thom class of an equivariant topological cycle of codimension twice the codimension of its fixed points (inside the fixed point set of ), these topological cycles will give rise to a cohomology frame.

     

A surprising covering of the real line

We construct an increasing sequence of Borel subsets of , such that their union is , but cannot be covered with countably many translations of one set. The proof uses a random method.