In the present paper, we introduce a general notion of quotient ring which is based on inverses a... more In the present paper, we introduce a general notion of quotient ring which is based on inverses along an element. We show that, on the one hand, this notion encompasses quotient rings constructed using various generalized inverses. On the other hand, such quotient rings can be viewed as Fountain-Gould quotient rings with respect to appropriate subsets. We also investigate the connection between partial order relations on a ring and on its ring of quotients.
COROLLARY 2. If the Space X admits a generalized ~0-representation and every space Xq, r has a ne... more COROLLARY 2. If the Space X admits a generalized ~0-representation and every space Xq, r has a net of type C, then X has a net of type C. Thus the class of spaces having a net of type C is stable under passage to spaces admitting a generalized ~o.-representation. The author expresses her deep appreciation to R. M. Makarov for his constant interest in this work and valuable editorial advice. 1o 2. LITERATURE CITED D. A. Raikov, "The closed graph theorem and its converse for linear topological spaces," Sib. Mat. Zh., 7, No. 2, 353-372 (1966). ~r de Wilde, ',Reseaux dans les espaces lindaires a seminormes,"
Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents ... more Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.
SummaryA *-primitive involution ring Ris either a left and right primitive ring or a certain subd... more SummaryA *-primitive involution ring Ris either a left and right primitive ring or a certain subdirect sum of a left primitive and a right primitive ring with involution exchanging the components. An example is given of a left and right primitive ring which admits no row and column finite matrix representation. We characterize *-primitive involution rings in terms of maximal *-biideals. A *-prime involution ring has a minimal left ideal if and only if it has a minimal *-biideal, and these involution rings are always *-primitive.
Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digi... more Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.
This paper presents a structural characterization of affine complete and locally affine complete ... more This paper presents a structural characterization of affine complete and locally affine complete semilattices.
On essential extensions, maximal essential extensions and iterated maximal essential extensions i... more On essential extensions, maximal essential extensions and iterated maximal essential extensions in radical theory, K.I. Beidar a structure theorem for Omega-groups, A.Buys quasi-division rings - some examples of quasi-regular rings, H.L. Chick strong hereditary strict radicals and quotient categories of commutative rings, B.J. Gardner radicals in Abelian groups, R. Goebel radicals of graded rings, E. Jespers a general approach to the structure of radicals in some ring constructions, A.V. Kelarev closed ideals in non-unital Morita rings, S. Kyuno radical extensions, W.G. Leavitt the distributive radical, R. Mazurek radical ideals of radically simple rings and their extensions, N.R. McConnell and T. Stokes classes of strongly semiprime rings, D.M. Olson, H.J. Le Roux and G.A.P. Heyman some questions concerning radicals of associative rings, E.R. Puczylowski some remarks about modularity of lattices of radicals of associative rings, E. Roszkowska some subidempotent radicals, A.D. Sands the radical of locally compact alternative and Jordan rings, A.M. Slin'ko on non-hypersolvable radicals of not necessarily associative rings, S. Tumurbat to the abstract theory of radicals - a contribution from near-rings, S. Veldsman complementary radical classes of proper semifields, H.J. Weinert and R. Weigandt.
Algebraic investigations of interpolation deal with two major problems. The first and obvious pro... more Algebraic investigations of interpolation deal with two major problems. The first and obvious problem is to find general conditions under which interpolation can be performed. The other consists in giving a more delicate description of the structure of algebras. The presence of the interpolation property implies that the algebra in question is simple. Simplicity of an algebra means actually that
A ubiquitous class of lattice ordered semigroups introduced by Bosbach, which we call Bezout mono... more A ubiquitous class of lattice ordered semigroups introduced by Bosbach, which we call Bezout monoids, seems to be the appropriate structure for the study of divisibility in various classical rings like GCD domains (including UFD's), rings of low dimension (including semi-hereditary rings), as well as certain subdirect products of such rings and certain factors of such subdirect products. A Bezout monoid is a commutative monoid S with 0 such that under the natural partial order (for a, b ∈ S, a ≤ b ∈ S ⇐⇒ bS ⊆ aS), S is a distributive lattice, multiplication is distributive over both meets and joins, and for any x, y ∈ S, if d = x ∧ y and dx 1 = x then there is a y1 ∈ S with dy1 = y and x1 ∧ y1 = 1. We investigate Bezout monoids by using filters and m-prime filters, and describe all homorphisms between them. We also prove analogues of the Pierce and the Grothendieck sheaf representations of rings for Bezout monoids. The question whether Bezout monoids describe divisibility in Bezout rings (rings whose finitely generated ideals are principal) is still open.
This article studies commutative orders, that is, commutative semigroups having a semigroup of qu... more This article studies commutative orders, that is, commutative semigroups having a semigroup of quotients. In a commutative order S, the square-cancellable elements S(S) constitute a well-behaved separable subsemigroup. Indeed, S(S) is also an order and has a maximum semigroup of quotients R, which is Clifford. We present a new characterisation of commutative orders in terms of semilattice decompositions of S(S) and families of ideals of S. We investigate the role of tensor products in constructing quotients, and show that all semigroups of quotients of S are homomorphic images of the tensor product R ⊗ S(S) S. By introducing the notions of generalised order and semigroup of generalised quotients, we show that if S has a semigroup of generalised quotients, then it has a greatest one. For this we determine those semilattice congruences on S(S) that are restrictions of congruences on S.
In the present paper, we introduce a general notion of quotient ring which is based on inverses a... more In the present paper, we introduce a general notion of quotient ring which is based on inverses along an element. We show that, on the one hand, this notion encompasses quotient rings constructed using various generalized inverses. On the other hand, such quotient rings can be viewed as Fountain-Gould quotient rings with respect to appropriate subsets. We also investigate the connection between partial order relations on a ring and on its ring of quotients.
COROLLARY 2. If the Space X admits a generalized ~0-representation and every space Xq, r has a ne... more COROLLARY 2. If the Space X admits a generalized ~0-representation and every space Xq, r has a net of type C, then X has a net of type C. Thus the class of spaces having a net of type C is stable under passage to spaces admitting a generalized ~o.-representation. The author expresses her deep appreciation to R. M. Makarov for his constant interest in this work and valuable editorial advice. 1o 2. LITERATURE CITED D. A. Raikov, "The closed graph theorem and its converse for linear topological spaces," Sib. Mat. Zh., 7, No. 2, 353-372 (1966). ~r de Wilde, ',Reseaux dans les espaces lindaires a seminormes,"
Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents ... more Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.
SummaryA *-primitive involution ring Ris either a left and right primitive ring or a certain subd... more SummaryA *-primitive involution ring Ris either a left and right primitive ring or a certain subdirect sum of a left primitive and a right primitive ring with involution exchanging the components. An example is given of a left and right primitive ring which admits no row and column finite matrix representation. We characterize *-primitive involution rings in terms of maximal *-biideals. A *-prime involution ring has a minimal left ideal if and only if it has a minimal *-biideal, and these involution rings are always *-primitive.
Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digi... more Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.
This paper presents a structural characterization of affine complete and locally affine complete ... more This paper presents a structural characterization of affine complete and locally affine complete semilattices.
On essential extensions, maximal essential extensions and iterated maximal essential extensions i... more On essential extensions, maximal essential extensions and iterated maximal essential extensions in radical theory, K.I. Beidar a structure theorem for Omega-groups, A.Buys quasi-division rings - some examples of quasi-regular rings, H.L. Chick strong hereditary strict radicals and quotient categories of commutative rings, B.J. Gardner radicals in Abelian groups, R. Goebel radicals of graded rings, E. Jespers a general approach to the structure of radicals in some ring constructions, A.V. Kelarev closed ideals in non-unital Morita rings, S. Kyuno radical extensions, W.G. Leavitt the distributive radical, R. Mazurek radical ideals of radically simple rings and their extensions, N.R. McConnell and T. Stokes classes of strongly semiprime rings, D.M. Olson, H.J. Le Roux and G.A.P. Heyman some questions concerning radicals of associative rings, E.R. Puczylowski some remarks about modularity of lattices of radicals of associative rings, E. Roszkowska some subidempotent radicals, A.D. Sands the radical of locally compact alternative and Jordan rings, A.M. Slin'ko on non-hypersolvable radicals of not necessarily associative rings, S. Tumurbat to the abstract theory of radicals - a contribution from near-rings, S. Veldsman complementary radical classes of proper semifields, H.J. Weinert and R. Weigandt.
Algebraic investigations of interpolation deal with two major problems. The first and obvious pro... more Algebraic investigations of interpolation deal with two major problems. The first and obvious problem is to find general conditions under which interpolation can be performed. The other consists in giving a more delicate description of the structure of algebras. The presence of the interpolation property implies that the algebra in question is simple. Simplicity of an algebra means actually that
A ubiquitous class of lattice ordered semigroups introduced by Bosbach, which we call Bezout mono... more A ubiquitous class of lattice ordered semigroups introduced by Bosbach, which we call Bezout monoids, seems to be the appropriate structure for the study of divisibility in various classical rings like GCD domains (including UFD's), rings of low dimension (including semi-hereditary rings), as well as certain subdirect products of such rings and certain factors of such subdirect products. A Bezout monoid is a commutative monoid S with 0 such that under the natural partial order (for a, b ∈ S, a ≤ b ∈ S ⇐⇒ bS ⊆ aS), S is a distributive lattice, multiplication is distributive over both meets and joins, and for any x, y ∈ S, if d = x ∧ y and dx 1 = x then there is a y1 ∈ S with dy1 = y and x1 ∧ y1 = 1. We investigate Bezout monoids by using filters and m-prime filters, and describe all homorphisms between them. We also prove analogues of the Pierce and the Grothendieck sheaf representations of rings for Bezout monoids. The question whether Bezout monoids describe divisibility in Bezout rings (rings whose finitely generated ideals are principal) is still open.
This article studies commutative orders, that is, commutative semigroups having a semigroup of qu... more This article studies commutative orders, that is, commutative semigroups having a semigroup of quotients. In a commutative order S, the square-cancellable elements S(S) constitute a well-behaved separable subsemigroup. Indeed, S(S) is also an order and has a maximum semigroup of quotients R, which is Clifford. We present a new characterisation of commutative orders in terms of semilattice decompositions of S(S) and families of ideals of S. We investigate the role of tensor products in constructing quotients, and show that all semigroups of quotients of S are homomorphic images of the tensor product R ⊗ S(S) S. By introducing the notions of generalised order and semigroup of generalised quotients, we show that if S has a semigroup of generalised quotients, then it has a greatest one. For this we determine those semilattice congruences on S(S) that are restrictions of congruences on S.
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