Showing papers on "Unit (ring theory) published in 1976"
TL;DR: In this article, it was shown that the category of right cancellative monoids and permissible homomorphisms is naturally equivalent to a category of O-simple inverse semigroups.
51 citations
TL;DR: In this paper, it was shown that there is a split exact sequence 1 -+ R(R) + BM(R, H) + Gal(H) + 1, where H is a projective, commutative, and cocommutative Hopf algebra over R. This sequence generalizes that obtained in [5] for graded algebras, the dual of the group ring RG.
45 citations
TL;DR: In this article, it was shown that the ring H ∞ of bounded analytic functions in the unit disc is coherent, while the disc algebra A is not coherent, for any positive measure μ, L∞(μ) is coherent.
34 citations
1 Jan 1976
TL;DR: In this article, it was shown that the divisor class group of RG is a homomorphic image of an extension of a subgroup of R by a subquotient of the character group of G.
Abstract: Let R be a normal affine domain over the algebraically closed field k, and let G be a connected algebraic group acting rationally on R. It is shown that the divisor class group of RG is a homomorphic image of an extension of a subgroup of the class group of R by a subquotient of the character group of G. In particular, if R has finitely generated class group, so does RG. The object of this note is to establish the following theorem: Let R be a normal affine domain over the algebraically closed field k, and let G be a connected algebraic group acting rationally on R. Then if R has a finitely generated divisor class group, then so does RG. (If K is the quotient field of R, then RG is R n KG, so RG is a Krull domain and hence has a divisor class group.) The following conventions are adopted: k is the fixed algebraically closed base field. For a commutative k-algebra A, U(A) denoLes the group of units of A and Uk(A) = U(A)/k*. We begin with some observations regarding group actions and units. PROPOSITION 1. Let R be an integral domain k-algebra with quotient field K suvh that Uk (R) is a finitely generated group, and let G be a connected algebraic group acting as k-algebra automorphisms of R, such that every unipotent subgroup of G acts rationally on R. Then: (a) Every f in U(R) is a semi-invariant for G. (b) If f is in K such that g(f )/f e U(R) for all g e G, then f is a semiinvariant for G. PROOF. First we consider the case where G is unipotent and R is the coordinate ring of the affine k-variety V. If f is a nonvanishing function on V and v an element of V, then g -> f (gv) is a nonvanishing function on G, hence constant since G is unipotent. Thus f is an invariant. In general R is a direct limit of such coordinate rings, and hence every unit of R is invariant under every unipotent subgroup of G. Now we can establish (a). We need to know that G acts trivially on Uk(R), and by the above paragraph it is enough to treat the case G = Gm. Now Uk (R) is a finitely generated free abelian group, and the action of Received by the editors December 29, 1975. AMS (MOS) subject classifications (1970). Primary 13A05; Secondary 20G 15. Copyright CD 1977, American Mathematical Society
30 citations
1 Mar 1976
TL;DR: In this paper, the authors introduce the concept of train algebras and define a rank equation in which the coefficients of a general element x depend only on its baric value, generally called the weight of x.
Abstract: Train algebras were first introduced by Etherington in ( 1 ) and proved very useful in dealing with problems in mathematical genetics. The types of algebras which arose were commutative, non-associative and finite-dimensional. It proved convenient in the general theory to regard them as defined over the complex numbers. We remind the reader of some basic definitions. A baric algebra is one which admits a non-trivial homomorphism into its coefficient field K . A (principal) train algebra is baric and has a rank equation in which the coefficients of a general element x depend only on its baric value, generally called the weight of x . A special train algebra (STA) is a baric algebra in which the nilideal is nilpotent and all its right powers are ideals; the nilideal being the set of elements of A of weight zero. In ( 2 ) Etherington showed that in a baric algebra one can always take a very simple basis consisting of a distinguished element of unit weight and all other basis elements of weight zero.
21 citations
TL;DR: In this paper, it was shown that a commutative integral domain R satisfies the property that whenever nonzero elements a and b in R are such that each element in the sequence a, b2, a3, b4, … divides the next, then they are associates in R (that is, a = bu for some unit u of R).
Abstract: Our starting point is an observation in elementary number theory [10, Exercise 26, p. 17]: if a and b are positive integers such that each number in the sequence a, b2, a3, b4, … divides the next, then a = b. Its proof depends only on Z being a unique factorization domain (UFD) whose units are 1, —1. Accordingly, we abstract and say that a (commutative integral) domain R satisfies (*) in case, whenever nonzero elements a and b in R are such that each element in the sequence a, b2, a3, b4, … divides the next, then a and b are associates in R (that is, a = bu for some unit u of R). The main objective of this paper is the study of the class of domains satisfying (*).
17 citations
TL;DR: The study of the nilpotency of U(RG) has been the subject of several papers, e.g., the authors, where the group ring of the group G over R and the group of units of this group ring were studied.
Abstract: Let R be a ring with unit element and G a finite group. We denote by RG the group ring of the group G over R and by U(RG) the group of units of this group ring. The study of the nilpotency of U(RG) has been the subject of several papers.
15 citations
1 Feb 1976
TL;DR: In this article, it was shown that U(KG) is solvable and n-Engel if and only if the derived group G' is a finite p-group.
Abstract: Let KG be the group ring of a group G over a field of characteristic p > 0, p =# 2, 3. Suppose G contains no element of order p (if p > 0). Group algebras KG with unit group U(KG) solvable and n-Engel are characterized. Let KG be the group ring of a group G over a field K of characteristic p > 0 and let U(KG) denote its group of units. Several authors including Bateman [1], Bateman and Coleman [2], Motose and Tominaga [10] and Khripta [5] have studied the question as to when U(KG) is solvable or nilpotent. Khripta in a beautiful paper [5] has proved that if p > 0 and G has a p-element then U(KG) is nilpotent if and only if G is nilpotent and the derived group G' is a finite p-group, settling the nonsemiprime case. This, incidently, is equivalent to saying that KG is Lie nilpotent (see [11] and [14]). Khripta also has some results in her thesis on the nilpotency of U(KG) in the semiprime case. We investigate when U(KG) is a solvable n-Engel group; more precisely we prove THEOREM. Suppose KG is a group ring over a field K of characteristic p > 0, p #2, 3. Suppose G has no element of order p (if p > 0). Then the following are equivalent. (i) U(KG) is solvable and n-Engel. (ii) G is solvable and m-Engel and one of (a), (b) holds. (a) T(G), the set of torsion elements of G, is central in G. (b) IKI = 2A 1 = p, a Mersenne prime; T(G) is abelian of exponent (p2 1) andfor x E G, t E T(G), xt #: tx =X xx tx= tP. (iii) U(KG) is nilpotent. We are indebted to the referee for several useful comments. 1. Notations and definitions. For group elements x, y we write the commutator (x, y) = xyxI 1 and (x, y,y, ...,y = (vxI, y. ..,y )y(x, y,...,y) y-' n n n+ 1nn A group H is n-Engel if it satisfies (x, y,...,y)= 1 forallx,y E H
11 citations
10 citations
TL;DR: In this paper, the finest inverse semigroup congruence on a semigroup S is given by a V b V(a) = V(b) where V denotes the set of inverses of a e S; and Y is 'idempotentdetermined' in that if b is idempotent and a Y b then a is idemepotent.
Abstract: An element u of a semigroup S is called a midd~.e unit if xuy = xy for all x,y e S. In what follows, we shall restrict our attention to the case where S is an orthodox semigroup. We denote by Y the finest inverse semigroup congruence on S. We recall [2] that Y is given by a V b V(a) = V(b) where V(a) denotes the set of inverses of a e S; and that Y is 'idempotentdetermined' in that if b is idempotent and a Y b then a is idempotent.
9 citations
1 Jan 1976
TL;DR: In this article, it was shown that the radical of the rational group ring cannot be too large and therefore the group ring G[G] is not algebraic over the rationals.
Abstract: It is shown that if F is a field of characteristic zero and G is a group such that the group ring F[G] is semilocal then G must be finite. A generalization to group rings over rings is given. A ring R is semilocal if R/J(R) is artinian, where J(R) denotes the Jacobson radical of R. R is said to be local if R/J(R) is a division ring. It is well known that the group ring is never local for a field of characteristic zero unless the group is trivial. As it is conjectured that J(F[G]) = (0) whenever F is a field of characteristic zero, we expect F[G] semilocal to imply G finite, and this is easily proved if F is not algebraic over the rationals, by a theorem of Amitsur [1]. The result in this paper may be interpreted as saying that the radical of the rational group ring cannot be "too large". We would like to thank D. S. Passman for his helpful suggestions. LEMMA 1. Let K be a central subfield of a division ring D and let T E Mn (D), the full n by n matrix ring over D. Then the set S(T) = {k E K: 1k' T is singular} has at most n elements. PROOF. When written on the right, the elements of Mn(D) may be regarded as left D-linear transformations from the vector space Dn to Dn. If 1 k T is singular, there is a nonzero vector v E Dn such that v(l k'T) = 0, or vT = kv since k commutes with v. Hence k is an eigenvalue of T. Standard arguments of linear algebra show that eigenvectors in Dn corresponding to distinct eigenvalues of T in the centre of D are D-linearly independent. COROLLARY. Let R be a completely reducible K-algebra and let x E R. Then the set S(x) = {k E K: 1 k-x is not a unit in R} is finite. The following clever lemma forms a major part of the proof of the theorem in [3]. LEMMA 2 (FORMANEK). Let K be a subfield of the reals and let x = I_ aigi Received by the editors February 23, 1976. AMS (MOS) subject classifications (1970). Primary 16A26, 16A46; Secondary 16A10.
TL;DR: In this article, the structure of the completion of a regular ring R with respect to a pseudo-rank function was studied and the connections between the ring-theoretic structure of such a completion and the geometric structure of a compact convex set lP(R) of all pseudo-Rank functions on R were established.
Abstract: This paper is concerned with the structure of the completion of a (von Neumann) regular ring R with respect to a pseudo-rank function, and with the connections between the ring-theoretic structure of such a completion and the geometric structure of the compact convex set lP(R) of all pseudo-rank functions on R. Given a finite or infinite positive convex combination N = 2;~ RPR in 1P(R), it is shown that if the Pk lie in pairwise disjoint faces of IP(R), then the N-completion of R is naturally isomorphic to the direct product of the Pk-completions of R. Given N, Pc IP(R), it is shown that P extends to a pseudo-rank function on the N-completion of R if and only if P lies in the closure of the face generated by N in IP(R). Other results develop representations of the faces and extreme points in IP(R). This research was partially supported by Grant No. GP-43029 of the National Science Foundation (USA). All rings in this paper are associative with unit, and ring maps are assumed to preserve the unit.
TL;DR: In this article, the associative and commutative laws are characterized by preservation under the construction of powers and addition of a (new) unit element, which is used to generate the varieties defined by the two laws from two element groupoids.
Abstract: The associative and the commutative laws are characterized by preservation under the construction of powers and addition of a (new) unit element. This is used to generate the varieties defined by the two laws from two element groupoids.
TL;DR: In this paper, it was shown that if the requirement that be algebraic is weakened to the demand that the projections onto the coordinate planes be open, then the conjecture holds and the proof uses only the openness of the projection.
Abstract: Let be an algebraic set in containing the origin and let be the unit sphere.CONJECTURE. The diameter of one of the connected components of is greater than one.In this article it is shown that this is false if the requirement that be algebraic is weakened to the demand that the projections onto the coordinate planes be open. If, however, is replaced by the boundary of the unit polydisc, then the conjecture holds and the proof uses only the openness of the projection.Bibliography: 3 titles.
TL;DR: In this article, the general solution of three functional equations occuring in interpolation theory are given in the general situation when variables belong to a semi-group with unit and some regular properties like continuity or measurability.
Abstract: In this paper, the general solution of three functional equations occuring in interpolation theory are given in the general situation when variables belong to a semi-group with unit. We then specialize to some particular cases by specifying either the semi-group, or adding some regular properties like continuity or measurability.
1 Jan 1976
TL;DR: Theorem 2 in this article states that a left Noetherian fully left bounded classical ring is left balanced, and it is possible to apply the results of [3] to the theory of Azumaya algebras.
Abstract: Theorem 2 in this note states that a left Noetherian fully left bounded classical ring is left balanced. Since such a ring is evidently also left seminoetherian, it is possible to apply the results of [3]. Further application to the theory of Azumaya algebras yields a generalization of the sheaf theory, as described in [7], [9], in that the structure sheaf on spec R is described acurately, even if the ring R is not prime. Also, localization theory of Azymaya algebras, included as an example in Section VI of [9], may be considerably simplified. R will be an associative ring with unit. The set of idempotent kernel functors in the category of left R-modules, M(R), will be denoted by P(R). Adapting notation from [2], R-sp will be the set of prime kernel functors in P(R). Recall that CJ E F(R) is symmetric if and only if the associated localizing filter T’(a) allows a filterbasis of a-dense ideals, of. [7]. R is said to be classical if and only if for every P E spec R, the multiplicative set Q(P)=(xER, yx~P if and only if ~/EP)
TL;DR: In this article, it was shown that a local ring having an elementary abelian group of units has characteristic two, four or eight and is a homomorphic image of Z k G/E(Z k G) where G is some elementary 2-group and E is the ideal generated by {1 - u 2:u∈(Zk G)*}.
Abstract: In [2] the structure of all semiperfect rings with abelian group of units has been obtained in terms of commutative local rings. It follows easily that the structure of semiperfect rings with elementary abelian group of units is determined by commutative local rings whose unit groups are elementary abelian. In this note such local rings are completely characterized. It is shown that a local ring having an elementary abelian group of units has characteristic two, four or eight and is a homomorphic image of Z k G/E(Z k G) where G is some elementary 2-group and E(Z k G) is the ideal of Z k G generated by {1 - u 2:u∈(Z k G)*}.
TL;DR: In this paper, the rank of a prime element P of a Noether lattice L is defined to be the supremum of all integers n for which there exists a prime chain Po
Abstract: For terminology used in the remainder of this paper which is not defined here, the reader is directed to the reterences. Most of the basic concepts can be found in [1]. Let A be an arbitrary element of a Noether lattice L. Rad (A) is defined to be the join of all elements X of L for which there exists a natural number n such that X " ~ A. Clearly A ~ Rad (A), and since L satisfies the ascending chain condition, for each element A of L there exists a natural number m such that (Rad (A))~<=A. By a maximal element of L we shall mean a maximal element different from the unit element of L. The rank of a prime element P of a Noether lattice L is defined to be the supremum of all integers n for which there exists a prime chain Po
TL;DR: In this paper, the lattice of principal right ideals of a regular ring R is characterized, and it is shown that R is unit regular if and only if complements of isomorphic summands of M are equivalent with respect to the relation "is isomorphic to a submodule of".
Abstract: A ring R is unit regular if for every a E R, there is a unit x E R such that axa = a, and one-sided unit regular if for every a E R, there is a right or left invertible element x E R such that axa = a. In this paper, unit regularity and one-sided unit regularity are characterized within the lattice of principal right ideals of a regular ring R (Theorem 3). If M is an A-module and R = EndA M is a regular ring, then R is unit regular if and only if complements of isomorphic summands of M are isomorphic, and R is one-sided unit regular if and only if complements of isomorphic summands of M are comparable with respect to the relation "is isomorphic to a submodule of" (Theorem 2). A class of modules is given for whose endomorphism rings it is the case that regularity in conjunction with von Neumann finiteness is equivalent to unit regularity. This class includes all abelian torsion groups and all nonreduced abelian groups with regular endomorphism rings. In [1], a ring R with identity was defined to be unit regular if for every a E R there is a unit x E R such that axa = a. The class of all unit regular rings includes [1] all semisimple Artinian rings, all continuous von Neumann rings [7], all strongly regular rings (in particular, all commutative regular rings [3] ). Using the characterization (cf. [6, p. 117] ) of regular group rings as the group rings of locally finite groups, it is easy to show that all regular group rings are unit regular. Unit regular rings are von Neumann finite [4, Proposition 1] and are elementary divisor rings [4, Theorem 3]. Every element of a unit regular ring in which 2 is a unit is equal to the sum of two units [1, Theorem 6]. An example of a regular ring which is not unit regular is the endomorphism ring of an infinite dimensional vector space [1]. In this paper, we define one-sided unit regularity and characterize both unit regularity and one-sided unit regularity within the lattice of principal right ideals of the ring (Theorem 3). For an A-module M with regular endomorphism ring R we prove that R is unit regular if and only if any two isomorphic complemented Presented to the Society, March 19, 1974; received by the editors April 4, 1974 and, in revised form, September 9, 1974. AMS (MOS) subject classifications (1970). Primary 16A30; Secondary 16A42, 16A48, 16A64, 20K30.
1 Jan 1976
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