Nonzero Intersection Research Articles

Applications of the Connes spectrum to C∗-dynamical systems, II

We show that for a C ∗-dynamical system ( A, G, α) with G discrete (abelian) the Connes spectrum Γ(α) is equal to G ̂ if and only if every nonzero closed ideal in G × α A has a nonzero intersection with A. Denote by G J the closed subgroup of G that leaves fixed the primitive ideal J of A. We show for a general group G that if all isotropy groups G J are discrete, then G X α A is simple if and only if A is G-simple and Γ( α) = G ̂ . This result is applicable not only when G is discrete but also when G≃ R or G≃ T provided that A is not primitive. Specializing to single automorphisms (i.e., G= Z ) we show that if (the transposed of) α acts freely on a dense set of points in A ̂ , then Λ(α)= T . The converse is only proved when A is of type I.

Applications of the Connes spectrum to C∗-dynamical systems

We show that the crossed product between a C ∗-algebra A and a locally compact abelian group G with a representation α as automorphisms of A is a prime C ∗-algebra (any two nonzero ideals have a nonzero intersection) if and only if two conditions are satisfied: (a) A is G-prime (any two nonzero G-invariant ideals of A have a nonzero intersection); and (b) the Connes spectrum Γ(α) of α equals the whole dual group of G.

Subdirect sum decompositions of endomorphism rings

In [4] Levy defined irredundant subdirect sums of rings. This provided a unique decomposition of a left quotient semisimple ring as a subdirect sum of left quotient simple rings. He also showed that irredundant subdirect sums of prime rings possess some rather strong uniqueness properties. In [2] essential subdirect sums were defined and Levy's decomposition theorem was obtained for essential subdirect sums. Furthermore in [3] it was shown that a ring R has a maximal left quotient ring which is a direct product of full linear rings if and only if R is an essential subdirect sum of rings whose maximal left quotient ring is full linear. In §1 of this paper we show that for subdirect sums of prime rings, essential and irredundant subdirect sums are identical. We also give some uniqueness properties of essential subdirect sums which are even stronger than those for irredundant subdirect sums. In §2 we show that if M is a torsionless 12-module and 22 is an essential subdirect sum of prime rings, then the endomorphism ring of M is also an essential subdirect sum of prime rings. We get this theorem as a consequence of a more general result about Morita contexts. The proof of the more general result is notationally more efficient. 1* Essential subdirect sums of prime rings* In what follows R will be an associative ring which is not assumed to have an identity. RM will express that M is a left JS-module. A submodule K of M is an essential submodule if every nonzero submodule of M has nonzero intersection with K. M will then be an essential extension

A note on weak Bezout rings

A weak Bezout ring is defined in [I] as an integral domain with unit (not necessarily commutative) such that any two principal right ideals with nonzero intersection have sum and intersection which are again principal. In this note we weaken the condition that a ring be a weak Bezout ring by means of the following theorem which enables us to drop the condition that intersections be principal. THEOREM. Let K be an integral domain in which each two principal right ideals with nonzero intersection have a principal sum. Then K is a weak Bezout ring.

Semisimple maximal quotient rings

Notation and Introduction. R denotes an arbitrary associative ring. A right R-module A over R will be denoted AR. BR is a large submodule of AR (AR is an essential extension of BR), if BR is a submodule of AR having nonzero intersection with every nonzero submodule of AR. A right ideal I of R is a large right ideal, if IR is a large submodule of RR. Given AR, Z(AR) is the singular submodule of AR [9], which consists of all those elements of A whose annihilators in R are large right ideals. Following Johnson [9], Q is a right quotient ring of R if Q is a ring with identity containing R as a subring (the identity of Q is the identity of R if R has one) and RR is a large submodule of QR. The quotient rings considered by Goldie in [6], [7] will be called classical quotient rings. Q is a classical right quotient of R if every regular element (nonzero divisor) of R is a unit in Q and every element of Q is of the form ab -1, a, b E R, b regular in R. In general, a ring R need not possess a classical right quotient ring. Goldie [7], has given necessary and sufficient conditions that a ring possess a classical right quotient ring which is semisimple. Here semisimple means semisimple with minimum condition [8]. This paper is concerned with the question of characterizing those rings which have a semisimple maximal right quotient ring [4], [9], [10], [11] and in this case generalizing some simple well-known results about commutative integral domains, their quotient rings and modules over these domains. Johnson [9] has shown that R has a regular maximal right quotient ring Q if and only if Z(RR) = 0, where Q is a regular ring [13] if every finitely generated right (left) ideal of Q is generated by an idempotent. In this case QR is injective [3] as a right R-module, hence the injective hull of R [2]. A ring R has a semisimple maximal right quotient ring Q if and only if Z(RR) =O and dim RR is finite, where a right R-module M is of finite dimension if every direct sum of submodules of M has only finitely many nonzero summands. This is the main result of ?1. In addition another characterization is given for rings which possess a semisimple classical right quotient ring, namely, R has a semisimple classical right quotient ring if and only if Z(RR) = 0 and if I is a large right ideal of R, then there is an element a E I such that aR is a large right ideal of R. If R has a semisimple classical right quotient ring Q, then it is known [3], that

On Very Large One Sided Ideals of a Ring

If R is a ring, a right (left) ideal of R is said to be large if it has non-zero intersection with each non-zero right (left) ideal of R [8]. If S is a set, let |S| be the cardinal number of S. We say a right (left) ideal I of a ring R is very large if |R/I| < < No. If a is an element of a ring R such that (a)r = {r ∊ R|ar = 0} is very large then we say a is very singular. The set of all very singular elements of a ring R is a two sided ideal of R. If R is a prime ring, then 0 is the only very singular element of R and a very large right (left) ideal of R is indeed large provided that R is not finite.

On Prime J-Rings with Uniform One-Sided Ideals

We say that a ring S is a Jl-ring if it fulfills (J1). A Jr-ring is defined in an obvious way. A ring S is called a left quotient ring (in the sense of R. E. Johnson) of a subring R if every nonzero left R-submodule of S has a nonzero intersection with R. A right quotient ring is defined similarly. An extension ring S of a ring R is called a two-sided quotient ring of R if S is a left quotient ring of R, and also a right quotient ring of R. For any J1-rinig S there exists such a left quotient ring Si that every left quotient ring of S has an isomorphic, over S, image in SI. Si is unique up to isomorphism over S, and is called the maximal left quotienit ring of S. Similarly the maximal right quotient ring Sr exists for any Jr-ring S. In case S is a J-ring, it may be shown that S has the maximal two-sided quotient ring St. A ring S is called a continuous transformation ring if there is a pair of dual vector spaces (V, V') such that S is the ring of continuous linear transformations of the vector space V topologized by the V'-topology. This concept is right-left symmetrical; that is, S may be regarded as the ring of continuous linear transformations of the vector space V' topologized by the V-topology. (See [2; ?? 6, 7, Chap. IV]). A left ideal A of a ring S is called uniform if we have B n C ? 0 for any nonzero left ideals B and C such that B, C C A. A uniform right ideal is defined similarly. The main theorem of this paper is the following: Let S be a prime J-ring

Rings with a weak algorithm

determinate over a field F. All these ideas can be generalized in a straightforward manner to the noncommutative case: the principal ideal domains again form a well-behaved though rather narrow class (cf. Jacobson [9, Chapter 3]), and the valuated rings with a Euclidean algorithm are just the skew polynomial rings k[x; S, D] over a skew field k with an automorphism S and an S-derivation D (2). One obtains a slightly larger class by taking, instead of principal ideal domains, Bezout rings, i.e., integral domains in which any finitely generated (left or right) ideal is principal but this probably amounts to not much more than allowing locally principal ideal domains. A significantly wider class of rings is obtained by taking all integral domains in which any two principal right ideals with a nonzero intersection have a sum and intersection which are again principal. These are the weak Bezout rings introduced in [6], where it is shown that a weak Bezout ring in which prime factorizations exist, is a unique factorization domain, and other decomposition theorems hold (corresponding to the primary decomposition of an ideal in a Noetherian ring). Further it is shown there that the weak Bezout rings include free associative algebras in any number of free generators over a field. It is possible to weaken the definition of the Euclidean algorithm in a similar way so as to obtain rings with a weak algorithm (cf. ?2 for the definition). This was first introduced in [4] where it was applied to prove (in effect) that in any ring R with a weak algorithm, all right ideals were free R-modules. We now continue the study of rings with a weak algorithm and in particular show that they are weak Bezout rings, so that the results of [6] become applicable (?4). This

On rings of which any one-sided quotient rings are two-sided

0. Introduction and preliminaries. Following R. E. Johnson [1] we assume in this paper that any rings we shall be concerned with satisfy either one or both of the following conditions: (Jf) If the right annihilator of a left ideal A is nonzero, then there exists a nonzero left ideal B such that A nB =0. (J) =the right left symmetry of (J1).I We say that a ring is a Ji-ring, a Jf-ring or a J-ring if it satisfies (Ja), (J,) or both of them. A module A is called an essential extension of a submodule B if BnCFCO for every nonzero submodule C of A. A module is said to be injective if it is a direct summand of every extension module. It is well known that every module M has a maximal essential extension ff. A is injective, and is unique to within an isomorphism over M. Let S be a Ji-ring. Then we can define the multiplication in the maximal essential extension 3 of the left S-module S such a way that (i) 3 forms a ring and (ii) the multiplication coincides, on SX3, with the scalar multiplication. This ring is unique up to an isomorphism over S, and is denoted by Sz. As is known, SI is regular (in the sense of von Neumann); and is left self injective, that is, injective as a left module over itself. An extension ring T of a Ji-ring S is called a left quotient ring of S if the left S-module T is an essential extension of the left S-module S. It is also known that every left quotient ring of S is isomorphic, over S, to a subring of 3;. Thus, Sl is the maximal left quotient ring of S. We define similarly a right quotient ring and the maximal right quotient ring ST of a Jr-ring S. For any J-ring S it is easily seen that the following conditions are equivalent: (i) There exists an extension ring T of S with the properties that (a) it is regular (both left and right) self injective, and (b) every nonzero one-sided S-submodule of T has a nonzero intersection with S. (ii) Every left quotient ring of S is a right quotient ring of S, and every right quotient ring of S is a left quotient ring of S. In this case any maximal left quotient ring of S and any maximal

Embedding numbers for finite groups

This note is concerned with the following problem. Let H denote a subgroup of a finite group G and let L denote a linear or one dimensional representation (i.e., a character) of H. We assume throughout that the field F is algebraically closed and is either of characteristic 0 or of prime characteristic which does not divide the order of any groups under consideration. Let GIL denote the corresponding induced representation of G. How many distinct (i.e., nonequivalent) irreducible representations appear in the decomposition of GI L into irreducible parts? (This number is just the central intertwining number of GI L, which is denoted by Ct(GI L). Cf. [1].) More specifically, we are interested in determining an upper bound on the number of distinct irreducible representations which will appear, purely in terms of the way H is embedded in G, and in terms which do not depend on the particular linear representation L of H. Two such bounds come quickly to mind. The number of classes (of conjugates) of the super group G, which we denote { G: e}, is clearly an upper bound. Dimension considerations also give [G: H] as an upper bound. We now introduce a new group theoretic invariant which heuristically is a measure of the manner in which the classes of G are distributed among the H-cosets of G. DEFINITION. Let H be a (not necessarily normal) subgroup of a finite group G. For each normal subset N of G, let +1(N) denote the number of classes (of conjugates) of G contained in N. Let +2(N) denote the number of right H-cosets of G which have nonzero intersection with N. Let +(N) = { G: e} -45(N) +42(N). We then define the embedding number of H in G, denoted by (G: H), to be the minimum of the +(N), as N is taken over all normal subsets of G. We remark that a definition of 42 using left cosets would yield the same value for (G: H) since N-' intersects the same number of left cosets as N does right cosets. Taking N= {e } where e is the identity element of the group we have (G: H) {G: e}. Taking N=G we have (G: H)? [G: H]. If H$ G, it is easy to verify that (G: H) > 1. If His a proper normal subgroup, then, taking N=H we have (G:H)<{G:e}. In the case where H is a normal subgroup of G, another number associated with the embedding of H in G is the number of classes in the factor group G/H. We call this the class number of H in G and denote it by { G: H}.

Hilbert property for double conic bundles and del Pezzo varieties

In this paper we prove that, over a Hilbertian ground field, surfaces with two conic fibrations whose fibres have non-zero intersection product have the Hilbert property. We then give an application of this result, namely the verification of the Hilbert property for certain del Pezzo varieties.