Abstract: The aim of this paper is to give a number of characterizations of the rings of the title. In particular, these turn out to be precisely those exchange rings whose idempotents are all central. They are also those rings in which every element is the sum of a unit and a central idempotent.

Abstract: We shall prove the following about the “ringification” ρA of [2] and [5] of an archimedean l-group A: (a) Any “minimal ring” containing A is ρA; (b) A ↦ ρA is a reflector; (c) ρA need not be laterally complete when A is. These constitute the solutions to the problems posed in [2] by Paul Conrad. 1. The embedding into a ring. Let be the category which has objects archimedean l-groups A with distinguished positive weak unit eA , and morphisms l-group homomorphisms h: A → B with h(eA) = eB . Let be the category with objects archimedean f-rings R with identity 1 R which is a weak unit, and morphisms l-ring homomorphisms h: R → S with h(l R ) = 1 S .

Abstract: Let n be a positive integer and let R be a prime ring either of characteristic zero or of characteristic > n. Then for any al, a2, . . ., a,+I E R, if aIxa2x *.* anxa +I = O for all x E R. Then ai = O for some 1 n. Then for any a1, a2, . .. ., an+1 E R, if alxa2x ... anxan+ I = O for all x E R, then ai = O for some I < i < n + 1. It should be noted that the theorem may not be true if the restriction on the characteristic of R is removed. Let R be the prime ring of m x m matrices over a finite field. We wish to show that for any x e R, there exists a nonzero ax ( R such that (aXx)2 = 0. First pick some nonzero a C R such that a2= 0. Then for x E R, if x is a unit let ax = ax-l and if x is not a unit, there exists a nonzero ax C R such that axx = 0. In any case ax # 0, and the equation HI (a,y)2=O x E R,x#0 holds for ally c R. We begin by linearizing the identity axa2x ... a,xan+l = 0. PROPOSITION 1. Let a1, a2, . . ., an+I be elements of a ring R. Suppose alxa2x... anxan + = Ofor all x C R. Then for all x, x2, ..., xn e R, aESn~~~~~~~~~~~~~~ E ax,(I)2X,(2 aXa()a+1= 0 where Sn is the symmetric group of degree n. PROOF. For J c {1, 2, . .. , n}, by 2Jalxk,a2xk2* anxkan+l we mean the sum over all indices k,, k2, . . , kn such that {kl, k2, . . . , kn} = J. Observe that for any integer 1 < j < n and for any xl, x2, .. ., xj C R, Received by the editors September 25, 1978. AMS (MOS) subject classifications (1970). Primary 16A12. ? 1979 American Mathematical Society 0002-9939/79/0000-0456/$02.25 27 This content downloaded from 207.46.13.71 on Fri, 21 Oct 2016 04:52:55 UTC All use subject to http://about.jstor.org/terms

Abstract: Let B be a ring, G a finite group of automorphisms acting on B and Bc the fixed subring of B. We give an example of a S which is quasi-Frobenius (Q-F) such that B c is not quasi-Frobenius. S. Jondrup (3) claims that if card G is a unit in B and if B is self-injective then BG is self-injective. J. Fisher and J. Osterburg (2) use this assertion to prove that if B is quasi-Frobenius then BG is quasi-Frobenius. However we show that this result fails. Suppose that A is a commutative local artinian ring with Jacobson radical R and call E the injective hull of the simple /I-module S = A/R. E is finitely generated and faithful (5, theoreme 2, p. 97 and corollaire 6, p. 99). B denotes the ring constructed on the abelian group A X E with the multiplication (a, e)(a', e') = (aa', ae' + a'e). Lemma. B is a commutative local quasi-Frobenius ring. It is clear that B is commutative local artinian with radical R X E and that S' = O X S is a simple ideal of B essential in the ideal O X E. O X E is essential in B, since for each a E A - (o) there exists e' E E such that (o, e')(a, e) = (o, ae') is nonzero, because E is faithful. Thus S' is essential in

Abstract: We consider a random walk Z, on the locally compact second countable abelian semigroup S with unit e which is assumed to be a recurrent point for Zn. Then S is the disjoint union of a topologically simple abelian semigroup G and a null-set A of first category. Under some additional conditions G is a topological group. Let S be a multiplicatively written locally compact, noncompact, second countable abelian semigroup with unit e and let ,u be a regular probability measure the support of which generates S. Let { Zn ln = 0, 1, 2, ... } be the random walk on S generated by ,u. Our basic assumption is that e is recurrent, i.e. Pr{Zn E UinfinitelyoftenIZ0 = e} = 1 (1) for every neighborhood U of e. We will show that (1) implies that S is the disjoint union of G and A where G is a topologically simple semigroup, i.e., G has no proper closed ideals, and A is a set of first category. Put v = 212-10u where ji" is the n-fold convolution product of ,u with itself. The support of v is all of S. Write x -> y if y can be reached from x, i.e. (x -'N) > 0 for every neighborhood N of y; here xIN-{s E SIxs E N). LEMMA. Let A be the set {y E Sly -i+ e}. Then A is a first category v-null-set. PROOF. Let { Un} be a neighborhood basis atie. Then A is the union of closed sets An-{y E SIv(y 'Un) = 0). An is closed since v(y'-Un) is a lower semicontinuous function of y. y e An is equivalent to yS n Un =0 which shows that An is an absorbing set. As in [6, p. 142] one sees that vik(An) > 0 for some k implies that XkUk(A,c) converges. In particular, k1 n(U,) converges which contradicts (1). ((1) together with the joint continuity of the multiplication implies that x -> x for all x.) Thus vik(An) = 0 for all k whence v(An) = 0 and v(A) = 0. Since the support of v is all of S no v-null-set can have interior points. Hence A is of first category. Suppose first that A is not dense in S. Then A is closed and it is a maximal closed ideal in S. To see this, let U be an open set in A c. Then A C UC. Let Received by the editors February 8, 1978. AMS (MOS) subject classifications (1970). Primary 60B99, 60J15. i 1979 American Mathematical Society 0002-9939/79/0000-0275/$01 .75 111 This content downloaded from 157.55.39.106 on Mon, 25 Apr 2016 05:36:51 UTC All use subject to http://about.jstor.org/terms

Abstract: Let R be a commutative ring with identity, and let A be a finitely generated R -algebra with Jacobson radical N and center C . An R -inertial subalgebra of A is a R -separable subalgebra B with the property that B+N=A . Suppose A is separable over C and possesses a finite group G of R -automorphisms whose restriction to C is faithful with fixed ring R . If R is an inertial subalgebra of C , necessary and sufficient conditions for the existence of an R -inertial subalgebra of A are found when the order of G is a unit in R . Under these conditions, an R -inertial subalgebra B of A is characterized as being the fixed subring of a group of R -automorphisms of A . Moreover, A ⋍ B ⊗ R C . Analogous results are obtained when C has an R -inertial subalgebra S ⊃ R .

Abstract: It is shown that under certain hypotheses the following conjecture is correct: on a noncompact complete hypersurface in Euclidean space the two conditions below cannot hold simultaneously: (i) the sum of principal radii of curvature is bounded; (ii) the support function is uniformly continuous. I. The main result. Let IS be a noncompact C3 hypersurface in Euclidean space E"'l (m > 2), and r(u) the position vector of C; u = {ui}, i = 1, .. ., m, are the local coordinates on C. Assume that 5' is equipped with the metric induced from E ' . Suppose also that 5 is orientable and oriented. If it is not so then we pass to the universal covering of Is and then work on that covering. Under such circumstances there exists on 5 a C2 vector field of unit normals, and one may consider the Gauss map y: J , where l is the hypersphere of unit radius in E"' centered at the origin. By 5* we denote a set on I which contains the limits of all converging sequences of the form Y(Pk) where Pk is a sequence of points on 5' unbounded in the metric of 57. In this note we shall study the asymptotic behaviour of T. For that reason our further assumptions are related to an "infinite" part of C. Assume that 5 * # 0. Then we call a "leaving domain" any submanifold 57' of IS with the following properties: (a) there exists an open domain B on E with a boundary of class Ck (k > 2) such that B n 9* = 0, aB n f * # 0 and y is a diffeomorphism mapping 5' onto B; (b) for any sequence of points nk E B converging to a point from 5i* the sequence y -( (nk) is an unbounded sequence on 5. We also say that a leaving domain 5' is asymptotically regular if the support function h(u) = (r(u), n(u)), n E 5T', transplanted via y on B can be extended to B + aB as a continuous function and its restriction p(u) = h(u)IaB is a CI.x function, 0 < X < 1. Here (, ) means, as usual, the inner product in E"'m . Received by the editors December 5, 1978 and, in revised form, February 14, 1979. AMS (MOS) subject classifications (1970). Primary 53C40. ? 1979 American Mathematical Society 0002-9939/79/0000-0566/$02.00

Abstract: I. The group CIA(G). Let A be a non-trivial commutative ring with a unit, G a discrete group. By RePA(G) we shall denote the category of all the representations of G on finitely generated modules over A ("A,G-modules"). Mod(A) will be the category of all the finitely generated modules over A. F:RePA(G) ~ Mod(A) is the forgetful functor. Define CIA(G) as the group of all the automorphisms of F which commute with the tensor product operation. That is to say: aECIA(G) means that for every XEOb(RePA(G)) there is an A-automorphism ~X of FX, such that if ~:X ~ Y is a morphism between two objects in RePA(G) , then the following diagram is commutative:

Abstract: The study on the regression coefficient of extent of credit met by commercial bank was found to be highly significant at one per cent level. This indicates that an increase in the extent of credit met by commercial bank by one unit would lead to an increase in the extent of adoption of improved farm practices by 1.71 units, other things being equal. The autonomous level of adoption was to be 1.0654.

Abstract: The steradian should be treated as a base SI unit rather than as a dimensionless derived SI unit

Abstract: A study on the credit pattern of Nationalised bank to farmers revealed a positive and significant relationship between the credit need of farmers and the extent of credit met by Commercial bank. For one unit increase in the quantum of credit needed, the extent of credit met increased by 0.42 unit. All the beneficiaries needed credit for purchase of seeds. fertilizers. pesticides and to meet cultivation charges and their need was mot to the extent of 75 per cent by the Commercial bank.

Abstract: A Study to evaluate the impact of cultivation of high yiniding varieties on demand for labour, wage structure. pattern of labour use and productivity of labour revealed that cultivation of high yielding varieties 1. Covered 36.99 per cent of the gross crop area of the sample farms. II. Created additional demand of 30 man days of labour per hectare. Ill. Increased the share of on-farm labour (i.e. family labour and attached labour) in total IV. Thus for, helped reduction in under-employment more than unemployment. V. In creased per capita earning of attached labour marginally (Rs. 3.23 against Rs. 2.96). VI. Increased number of days of active employment (184 days as against 170 man days), VII. Reduced disparity in wages earned by attached labourers and casual labourers and there by increased stability of the system of contract employment that offered security of job to farm workers. VIII. Increased average value produce of labour (Rs.14.33 against Rs. 12.08 per man day unit) by 18.63 per cent.

Abstract: Preview this article: Teaching the Paragraph as a Structural Unit, Page 1 of 1 < Previous page | Next page > /docserver/preview/fulltext/ccc/30/2/collegecompositionandcommunication16238-1.gif

Abstract: Preview this article: Preliminary Notes for a Possible Nine-Week Unit on ";The Quest of the Fish in Literature", Page 1 of 1 < Previous page | Next page > /docserver/preview/fulltext/ej/68/7/englishjournal14104-1.gif