Search

In the passage from fields to rings of coefficients quadratic forms with invertible matrices lose their decisive role. It turns out that if all quadratic forms over a ring are diagonalizable, then in effect this is always a local principal ideal ring? with?. The problem of the construction of a `normal' diagonal form of a quadratic form over a ring? faces obstacles in the case of indices greater than?1. In the case of index?2 this problem has a solution given in Theorem?2.1 for (an extension of the law of inertia for real quadratic forms) and in Theorem?2.2 for containing an invertible non-square. Under the same conditions on a ring? with nilpotent maximal ideal the number of classes of projectively congruent quadratic forms of the projective space associated with a free -module of rank? is explicitly calculated (Proposition?3.2). Up to projectivities, the list of forms is presented for the projective plane over? and also (Theorem?3.3) over the local ring with non-principal maximal ideal, where is a field with an invertible non-square in? and . In the latter case the number of classes of non-diagonalizable quadratic forms of rank?0 depends on one's choice of the field? and is not even always finite; all the other forms make up 21 classes.

Let R be a commutative ring with identity and M be a free R-module then we always have a representation of R, that is homomorphism ring μ: R → End R (M), with μ(r) := μ r : M → M and μ r (m) = rm for all r ∈ R and for all m ∈ M. In this paper, we will present some properties of representations of ring R on R-module, based on some notions in representation of R on vector space, such as admissible submodule, equivalence of two representations, decomposable representation and completely reducible representation. It will be shown that if M, N are two free R-modules then two representations μ: R→ End R (M) and : R → End R (N) are equivalent if and only if there is a module isomorphism T : M → N. If R is a principle ideal domain(PID), then it will be shown that every submodule of M is an admissible submodule of M, every representation of ring R on a free R-module is decomposable, and a representation of R on M is completely reducible if and only if M is semisimple.

The article studies rings that are direct sums of an infinite family of subrings. The question of the completeness of these rings in maximal ring topologies is investigated for various classes of ring topologies. In particular it is shown that if the continuum hypothesis is assumed, then both complete and non-complete maximal ring topologies exist on a ring that is an infinite direct sum of copies of the same finite field.

The problem of describing the lengths of the independent cycles in the indicated substitution reduces to the case where the ring and characteristic polynomial of the corresponding matrix are primary. The concept of a distinguished polynomial over a primary (local) ring R is introduced and studied. These polynomials are used to obtain formulas for the cycle types of linear substitutions that generalize known formulas for the case where R is a field. If R is a principal ideal ring, the formulas are practically computable. In the case where R is a Galois ring, there are given a complete description of the linear substitutions of maximal order and an algorithm for enumerating the cycles in such substitutions. Estimates of the exponents of the full linear group over a local ring and its congruence subgroup are given.Bibliography: 17 titles.

Suppose is a division ring, and is a left-ordered group such that for any Dedekind cut of the linearly ordered set the group is such that is a right Ore domain and the group is cofinal in . Then the group ring can be embedded in a division ring having a valuation in the sense of Mathiak with values in . If is the group of a trifolium, this construction leads to an example of a chain domain with a prime, but not completely prime, ideal.Bibliography: 9 titles.

Reciprocity laws and Mennicke symbols for regular rings of arbitrary dimension are introduced and studied. The universal Mennicke symbol for the polynomial ring relative to the principal ideal is computed. The results obtained in this paper are applied to the problem of determining the precise value of the stable rank of polynomial rings and affine algebras over fields. Bibliography: 19 titles.

In this paper we give necessary and sufficient conditions allowing us to tell whether or not the ring of invariants of a Gorenstein ring is Gorenstein.Bibliography: 8 titles.

It is proved that for a complete discrete valuation ring of zero characteristic with residue field of positive characteristic and maximal ideal , the natural homomorphism of -groups with coefficients is an isomorphism for all positive and . For the ring of integers in a local field , the groups are finite. Bibliography: 13 titles.

Let be an associative ring, and a right ideal of , i.e. . An element is quasiregular relative to if for a suitable . A right ideal is quasiregular relative to if all the elements of are quasiregular relative to . If , , then put: Theorem 1. Let . Then the sum of all right ideals which are quasiregular relative to is itself quasiregular relative to . Moreover, is a maximal modular right ideal of?We say that is a primitive right ideal of the ring if there exists a maximal modular right ideal satisfying . If , then , and therefore the ring is primitive.Density Theorem. Let be a primitive right ideal of the ring , and any of the maximal modular right ideals corresponding to , i.e. . Consider the irreducible right -module as a linear space over the division ring . Then, for any nonempty finite linearly independent subset of the linear subspace , and any other subset of , there is always an element such that .It is not hard to see that for Theorem 1 turns into the well-known characterization of the Jacobson radical, and the density theorem turns into the usual Jacobson-Chevalley density theorem for primitive rings.Bibliography: 4 titles.

It is proved that there exist multiplicative structures in the symplectic bordism theories with singularities of types and , where and , and that the ring is isomorphic to a polynomial ring , where and .Bibliography: 10 titles.

This paper deals with one-dimensional (commutative) rings without nilpotent elements such that every ideal is generated by three elements. It is shown that in such rings the square of every ideal is invertible, i.e. divides its multiplier ring. In addition, every ideal is distinguished, in the sense that on localization at any maximal ideal it becomes either principal or dual to a principal ideal. Conversely, if a one-dimensional ring without nilpotent elements satisfies either of these conditions, and if all its residue class fields are 2-perfect and contain at least three elements, then every ideal can be generated by three elements.Bibliography: 18 titles.

Let be a ring, and the left and right annihilators of the element , the two-sided ideal in called the additive controller, and let be an -isomorphism (i.e., multiplicative isomorphism) and its defect. An ideal in the ring is called an -ideal if for all -isomorphisms , is an ideal in and if and only if . It is shown that Very general sufficient conditions are given that a multiplicative isomorphism of subsemigroups of multiplicative semigroups of rings be extendible to the isomorphism of the subrings generated by them. Minimal prime ideals and the prime radical of a ring are -ideals. The strongly regular and regular rings that have unique addition are characterized.Bibliography: 29 titles.

An ideal in a commutative ring is called subinjective if it is the homomorphic image of an injective module. It is proved that all ideals in a commutative ring are subinjective if and only if the ring is a direct sum of local rings with this property. Necessary and sufficient conditions are given for all ideals to be subinjective in the local case. In particular, this is the case for self-injective rings whose ideals are linearly ordered, and for local self-injective rings in which the maximal ideal has a nontrivial annihilator.Bibliography: 7 titles.

A number of authors have introduced ring groups as objects generalizing locally compact groups. An analogue of the Pontrjagin principle of duality holds for ring groups. In this paper we introduce a wider class of ring groups, one including the locally compact groups.A construction is given whereby to each ring group there is defined a dual ring group ; here . By definition a ring group is determined by a -algebra (the space of the ring group) equipped with an additional structure which allows to be considered, in particular, as a Hopf-von Neumann algebra. When is a locally compact group, is the -algebra of bounded measurable functions on , considered in the natural way as operators in .Bibliography: 15 items.

The main result of this paper is the following theorem: If is a Noetherian ring, then the canonical homomorphism is surjective when and injective when .Bibliography: 9 titles.

In this paper the author constructs a chain ring (i.e. a ring in which the right and left ideals are linearly ordered by inclusion) with the following properties: 1) is a prime ring; 2) the Jacobson radical of is a simple chain ring (without identity); 3) each element of is a right and left zero divisor. This example gives an answer to one of Brung's questions. In addition, the ring is totally singular, i.e. it coincides with its right (left) singular ideal.The construction is based on a theorem that permits one to assign a chain ring to a right ordered group whose group ring can be imbedded in a division ring.Bibliography: 9 titles.

The following theorem (a conjecture of Rourke) is proved: every commutative ring spectrum of characteristic 2 which has finite type is isomorphic to the spectrum of ordinary cohomology theory with coefficients in .Bibliography: 13 titles.

In this paper we study the concept of the invariance of a ring relative to a commutative semigroup. Invariance is proved for certain Prufer rings and affine algebras relative to semigroups of a special class.Bibliography: 11 titles.

Let be a finite extension of the field of rational -adic numbers, the ring of all integral elements of , the multiplicative group of , a finite group, and the crossed group ring of and with the factor system (; ). A classification is given of the rings for which the number of indecomposable -representations is finite. When is a group ring, this problem was solved in papers by Faddeev, Borevic, Gudivok, Jakobinski, and others.Bibliography: 22 titles.

A description is given of two-sided Noetherian rings over which all finitely generated modules are semichained. Bibliography: 31 titles.

This article contains a study of the matrix groups En(R), SLn,SL'n, GLn and GL'n over an arbitrary commutative ring R, for n ≥ 4. Bibliography: 24 titles.

Every such ring is a direct sum of matrix rings over finite completely primary principal ideal rings. These latter rings are called Galois-Eisenstein-Ore rings or GEO-rings. A number of defining properties for GEO-rings are given, from which it follows that a finite ring with identity in which every two-sided ideal is left principal is a principal ideal ring. A theorem on the existence of a distinguished basis in a fintie bimodule over a Galois ring is proved, generalizing a similar theorem of Raghavendran. Finally, a GEO-ring is described as the quotient ring of an Ore polynomial ring over a Galois ring by an ideal of a special form, generated by Eisenstein polynomials. Bibliography: 10 items.

Let be a ring and its Jacobson radical. Let us set , , and if is a limit ordinal. We call a ring an annihilating ring if the left (right) annihilator of the right (left) annihilator of an arbitrary left (right) ideal is itself. We prove that a ring is quasi-Frobenius if and only if it is a left self-injective annihilating ring and for some transfinite .Bibliography: 15 items.

Conditions are advanced in the paper which, imposed on a ring of holomorphic functions of several variables and a finite collection of functions in that ring, are sufficient for the whole ring to be generated as the closed ideal spanned by the functions in the collection.

Let be a semisimple -ring and its center. Assume that for any prime ideal the ring is primary. Let be the intersection of the maximal over-rings of , and . We prove that has a finite number of indecomposable integral representations if and only if is a hereditary ring, has two generators as a -module, and is cyclic.

The following theorem is proved. If a ring R is radical over a commutative subring K, then all the nilpotent elements of R generate a null-ideal T for which the corresponding factor ring is commutative. An affirmative answer is thus provided for a question raised by Faith. Bibliography: 5 items.