Abstract: Recently Buchsbaum and Eisenbud [3] exploited the algebra structure on a finite free resolution of a Gorenstein ideal of codimension three to obtain a complete determinantal description of such an ideal. As they pointed out, the study of algebra structures on resolutions has for the most part been confined to the (generally infinite) minimal free resolution of the residue field of a local ring or the Koszul resolution of an ideal generated by a regular sequence. They proposed, however, to extend the scope of the study to all minimal free resolutions of cyclic modules. Khinich [1] furnished an example of a grade four ideal I for which the minimal resolution of R/I does not admit the structure of an associative, differential, graded commutat ive algebra (DGC algebra). Khinich's ring R/I is Cohen-Macaulay, but not Gorenstein. We conjecture that minimal finite free resolutions of Gorenstein factor rings R/a admit D G C algebra structures. In this paper we establish the conjecture for R a Gorenstein local ring in which 2 is a unit and a a Gorenstein ideal of grade (or height) four. We begin by clarifying what we mean by an algebra structure on a resolution. Let

Abstract: It is proved that if u is an element of a faithful algebra over a commutative ring R, then u satisfies a polynomial over R which has unit content if and only if the extension R ⊂ R[u] has the imcomparability property. Applications include new proofs of results of Gilmer-Hoffmann and Papick, as well as a characterization of the P-extensions introduced by Gilmer and Hoffmann.

Abstract: If H is an abelian extension of an imaginary quadratic field K, then the group of units of H contains an important subgroup consisting of modular units. These are obtained as special values of certain modular functions, and are analogous to the cyclotomic units of an abelian extension of Q. Suppose that H is an abelian extension not only of K but of Q as well. In this case, H contains both cyclotomic units and modular units, and the question arises as to the relation of these two groups of units. The study of this situation was begun by Kubert-Lang [KL 1] in the case K = Q(1/-p), H = Q(p,) (if p- 3 mod 4), or Q(p,,) (if p _ 1 mod 4). In this paper, we consider the general case of an arbitrary abelian extension of Q containing an imaginary quadratic field. We show that, for some power 21 of 2, the 21th power of the modular unit group of H is contained in the cyclotomic unit group of H. The exponent X depends only on the imaginary quadratic field K. We do this by deriving explicit formulae expressing modular units as power products of cyclotomic units. In Section 1 we define the two unit groups with which we will be dealing. We also describe more explicitly the fields H which are simultaneously abelian over Q and K. In Section 2, we use a factorization of L-series to obtain a set of linear relations between the logarithms of the modular units of H and those of the cyclotomic units of H. In the next two sections, we show how to solve these relations to express the modular units in terms of the cyclotomic units. At this point, we have to consider two separate cases. We treat the first, which is much simpler, in Section 3, providing a pattern for the more involved case of Section 4. In Section 5 we show that the exponents occurring in the products derived in Sections 3 and 4 are actually integral. Following a suggestion of the referee, we have replaced our original proof, which was by direct calculation, and have instead deduced this from the fundamental Theorem

Abstract: It is well known that every right Bezout domain satisfying the left Ore (multiple) condition is a left Bezout domain. A similar statement for the smaller class of principal right ideal domains is a long-standing conjecture which remains unresolved. In this paper we settle the analogous question for the larger class of right LCM domains. This paper deals with the question of the left-right symmetry of a right LCM domain, i.e. an integral domain in which the intersection of two principal right ideals is again principal. In the smaller class of right Bezout domains it is known that these are left Bezout domains provided it is assumed that they satisfy the left Ore condition: Ra n Rb =# 0 for all nonzero a, b in R. This follows from the fact that the definition of a weak Bezout domain (also known as a 2-fir) is left-right symmetric [3]. In an even smaller class the question of whether a left Ore PRI (principal right ideal) domain is a PLI domain remains open. Below it is shown that a left Ore right LCM domain need not be a left LCM domain in general but will be under the additional hypothesis that the ring has the ascending chain condition for principal left ideals. In fact the example given is that of a left and right bounded right LCM domain which is not a left LCM domain. This stands in contrast to the corresponding result that a left bounded PRI domain is a PLI domain [2]. In what follows R denotes a ring with unity and without proper divisors of zero; R* denotes the monoid of nonzero members of R. For x E R* let [xR, R] be the set of principal right ideals of R that contain x. Note that each [xR, R] is partially ordered by inclusion. PROPOSITION 1. For each x in R* the intervals [xR, R] and [Rx, R] are dually isomorphic; in particular if x = aa' then the correspondence aR < Ra' is a biyection which reverses order. PROOF. Let x = aa' = bb'. Then aR C bR iff a = bc for some c in R, and this is so iff b' = ca' for some c in R, i.e. iff Rb' C Ra'. Note that either containment becomes equality iff c is a unit in R. PROPOSITION 2. If the interval [xR, R] is a lattice for each x in R* and if R is a right Ore domain then R is a right LCM domain. Received by the editors February 6, 1979. AMS (MOS) subject classifications (1970). Primary 16A02.

Abstract: McCarthy [41 showed that a polynomial ring over a commutative von Neumann regular ring is semihereditary. Camillo [11 proved the converse. In this paper we examine polynomial rings over von Neumann regular rings which are not necessarily commutative. By a ring R we shall mean an associative ring with unit element. R is von Neumann regular if for each a E R, there is an a' in R such that aa'a = a. Then e = aa' is an idempotent and aR = eR. A ring S is right (resp. left) semihereditary if each of its finitely generately right (resp. left) ideals is projective as an S-module. Let S = R[x] be the polynomial ring in the (commuting) indeterminate x. THEOREM. The following are equivalent. (a) R is von Neumann regular. (b) For each a E R, aS + xS is a projective right ideal of S. (c) For each a E R, Sa + Sx is a projective left ideal of S. PROOF. Let R be von Neumann regular. If a E R then there exists an a' E R satisfying aa'a = a. Let e = aa'. From the equations a = (e + (1-e)x)a, x =(e + (1-e)x)( -e + ex), e + (1 e)x = aa' + x(I e), we deduce that aS + xS = (e + (1 e)x)S. It is easily verified that f = e + (1 e)x is a regular element (= nonzero-divisor) of S so that left multiplication by f induces an isomorphism between S and fS. Hence aS + xS is projective, proving (a) X (b). Suppose that for all a E R, aS + xS is projective. Fix a E R and let K = aS + xS. By the dual basis lemma for projective modules (see for example [2, p. 141]) there exist S-homomorphisms a and /8 from K into S, such that for every k E K, k = aa(k) + x/3(k). In particular, a aa(a) = x43(a). Since x is central in S and a is an S-homomorphism, xa(a) = a(a)x = a(x)a so that ax aa(x)a = x2f3(a). Equating coefficients of x on both sides, we obtain a = aa'a where a' is the coefficient of x of the polynomial a(x). Hence R is von Neumann regular, proving (b) X (a). The equivalence of (a) and (c) now follows from the left-right symmetry of (a). Received by the editors February 26, 1979 and, in revised form, May 25, 1979. AMS (MOS) subject classifications (1970). Primary 13B25; Secondary 16A30.

Abstract: PURPOSE:To obtain a CRT display unit available for highspeed movement by providing a coordinate register which holds coordinates indicating the position of a CRT memory on a display screen and a coordinate comparator circuit which generates a signal when coordinate values exceed limit values. CONSTITUTION:Display information from electric computer 10 is inputted to CRT controller 20 through interface 11. The information is discriminated by display controller 21 and then inputted to CRT memory 22, coordinate register 28 or memory block converter circuit 26 in accordance with its contents. Register 28 indicates what part of memory 22 should be displayed on CRT display unit 30, and information on it is supplied from computer 11 and shift signal generating circuit 40 through shift signal control circuit 27. Chaniging coordinate vlues of register 28 by a command from circuit 40 provides movements of the contents on display unit 30 in any direction. Further, coordinate comparator circuit 29 makes checks on upper and lower limits of coordinate all the time and when a coordinate is detected exceeding the limit, an interruption signal is supplied to the computer and the screen is corrected in the movement direction by a command from the computer.

Abstract: Let S be a compact connected finite dimensional monoid whose group of units G is a compact connected Lie group. Then there is an open set W about the unit element such that any compact subgroup within W has dimension at most dim S dim G 1 and if any toral subgroup achieves this dimension then that toral subgroup lies in the centralizer of G. Two applications are given, one to embeddings of irreducible monoids into S. Let G denote the group of units of a compact connected monoid, say, with zero, and let Z(G, S) denote the centralizer of the group of units. The structure of Z(G, S) is basically unknown. Indeed, it is an unsettled conjecture that it is connected [3]. For this aspect of Z(G, S) see [2]. It is the purpose of this note to place certain elements and subgroups in Z(G, S). It will be shown that toral subgroups of maximal dimension sufficiently close to the unit element must lie in Z(G, S). This will be used to show that certain irreducible monoids must also lie in Z(G, S). Let G be a compact group which is the group of units of a finite dimensional compact connected monoid S. If B is a closed subgroup of S outside of the minimal ideal the product GB can have dimension at most dim S 1. (See [1].) If, in fact, one has dim GB = dim S 1 then GB is a left group. Thus, if BG also has dimension dim S 1 one can conclude that B meets Z(G, S) = the centralizer of G. For this and related items see [4]. Now given any open set V about the unit in a compact connected monoid there may exist nontrivial compact connected subgroups outside of the group of units. Indeed, using [1] and [3] one can show the following: Let S be a compact connected monoid which is not a group. Suppose there is an open set W in S such that W contains no connected subgroup outside of the group of units. Then there is a closed ideal J such that S/J contains a thread from zero to unit. The following lemma is hardly unknown. It is stated for convenience. If G is a transformation group of X, the stability subgroup at x is denoted by Gx. In the following lemma there is no need to distinguish left and right stability. LEMMA 1. Let G be a connected group of units of a comwact monoid and let 0 be an open set about the unit. There exists about G an open set W such that x E W implies Gx C 0. In particular, if G is finite dimensional, there is about G an open set V such Received by the editors April 14, 1978 and, in revised form, April 2, 1979. AMS (MOS) subject classifications (1970). Primary 22A15; Secondary 22C05, 22E20. i 1980 American Mathematical Society 0002-9939/80/0000-0223/$03.25

Abstract: Let A be a complex Banach algebra with unit 1 satisfying ∥1∥ = 1. An element u in A is said to be unitary if it is invertible and ∥u∥ = ∥u −1 ∥ = 1. An element h in A is said to be hermitian if ∥exp( ith )∥ = 1 for all real t ; that is, exp( ith ) is unitary for all real t . Suppose that J is a closed two-sided ideal and π: A → A/J is the quotient mapping. It is easy to see that if x in A is hermitian (resp. unitary), then so is π(x) in A/J . We consider the following general question which is the converse of the above statement: given a hermitian (resp. unitary) element y in A/J , can we find a hermitian (resp. unitary) element x in A such that π(x)=y? (The author has learned that this question, in a more restrictive form, was raised by F. F. Bonsall and that some special cases were investigated; see [1], [2].) In the present note, we give a partial answer to this question under the assumption that A is finite dimensional.

Abstract: Given an associative ring R with unit we construct the largest homomorphic image of R such that every right invertible square matrix is left invertible. The construction may be accomplished in one step. Related results are also given. In this note all rings are associative with unit element. Such a ring R is said to be weakly finite if, for any square matrices A and B over R such that AB = I (the identity matrix), we also must have BA = I. The terminology is due to P. M. Cohn ([1, 0.2], [21), but other authors have considered this condition under other names (for example Goodearl, in [31). If R is not weakly finite, form an ideal J by including the entries of all I BA where AB = I and A and B are square. Then R/J is closer to being weakly finite, but we may have to repeat the process (possibly infinitely) to obtain finally a weakly finite homomorphic image of R. The purpose of this note is to show that no such repetition is necessary; that is, R/J is itself weakly finite. Other results can be obtained by keeping track of the sizes of the matrices involved. Suppose that A and B are n x n matrices over R and that AB = I. We think of A as a map of free right R-modules, and get an exact sequence