Hilbert Rings Arising as Pullbacks

Let S be a cancellative monoid with quotient group of torsion-free rank a.We show that the monoid ring R[S] is a Hilbert ring if and only if the polynomial ring R[{ X, },s/] is a Hilbert ring, where |/| = a.Assume that R is a commutative unitary ring and G is an abelian group.The first research problem listed in [K, Chapter 7] is that of determining equivalent conditions in order that the group ring R[G] should be a Hilbert ring.Matsuda has considered this question in [Ml, 6].His results show that for G finitely generated, R[G] and R are simultaneously Hilbert rings; if G is not finitely generated, then R[G] a Hilbert ring implies that R is a Hilbert ring, but the converse fails.In [M2, 5] Matsuda considers briefly the corresponding Hilbert-ring-characterization problem for a monoid ring R[S], where S is torsion-free and cancellative.Let a be the torsion-free rank of G and let {Xj)lsl be a set of indeterminates over R of cardinality a.In Corollary 1 we show that R[G] and P[{^,},e/] are simultaneously Hilbert rings.While the problem of determining equivalent conditions for P[{Ar,}/e/] to be a Hilbert ring has not been completely resolved, it has been worked on extensively [Kr, Go, L, Gl, H], and a significant body of positive results exists concerning this problem.The proof of Theorem 1 suggests the following conjecture: if S is a cancellative commutative monoid with quotient group G, then R[S] is a Hilbert ring if and only if R[G] is a Hilbert ring; this result is established in Theorem 2.All monoids considered are assumed to be commutative, and rings are assumed to be commutative and unitary.The statement of Theorem 1 uses the following terminology.For a cardinal number a, an extension ring T of R is said to be a-generated over R if T is generated over R by a set of cardinality at most a.Theorem 1. Assume that X = {XA is a set of indeterminates of cardinality a over the ring R. Denote by Za the direct sum of a copies of the additive group Z of integers.The following conditions are equivalent.(1) R[X] is not a Hilbert ring.(2) There exists a prime ideal P of R such that R/P admits an a-generated extension ring that is a G-domain, but not afield.(3) The group ring R[Za] is not a Hilbert ring._

In this paper, we study the maximal ideals in a commutative ring of bicomplex numbers and then we describe the maximal ideals in a bicomplex algebra. We found that the kernel of a nonzero multiplicative -linear functional in a commutative bicomplex Banach algebra need not be a maximal ideal. Finally, we introduce the notion of bicomplex division algebra and generalize the Gelfand–Mazur theorem for the bicomplex division Banach algebra.

A necessary and sufficient condition is given for a semilocal domain to have the property that every finitely generated module is a direct sum of cyclic modules. Recent work by Tom Shores and Roger Wiegand [SW] and [SWB] has shed new light on the old question: What is the class of commutative rings for which every finitely generated module is a direct sum of cyclic submodules? Following their terminology, such a ring is called an FGC-ring. In this paper, we are able to settle one of their conjectures [SWB, p. 1279], thus giving a characterization of reduced FGC-rings with Noetherian maximal ideal spectrum. We show that, for R a reduced ring with Noetherian maximal ideal spectrum, R is an FGC-ring and only R is a finite direct sum of h-local Bezout domains and each localization of R is an almost maximal valuation ring. The if part of the statement above was shown in [SW] to be an easy consequence of results due to Eben Matlis [M]. Other known results on FGC-rings are reviewed in [SW]; the reader is also referred to [SW] for definitions of the concepts above. The examples constructed in [W] show that our main result answers the FGC question for a large class of rings where it was unknown. Of course it would be nice to give a complete characterization of reduced FGC-rings. This would require settling another question in [SW]: Does every FGC-ring have Noetherian maximal ideal spectrum? They show the answer is yes the ring has fewer than 2' prime ideals [SW, Corollary 6.8]. 1. The main theorem. The result stated in the introduction will easily follow from the Theorem below: Theorem. Let R be an FGC-domain. Then every nonzero prime ideal P is contained in a unique maximal ideal. Proof. Suppose R contains two distinct maximal ideals M and N with Received by the editors May 16, 1974. AMS (MOS) subject classifications (1970). Primary 13C05, 13F05; Secondary 13A15. Copyright ? 1975, Anierican Mathematical Society

The purpose of this paper is to introduce two new classes of rings that is closely related to the classes of piecewise Noetherian domains and piecewise w-Noetherian domains. Let is a commutative ring and is a divided prime ideal of Let be a ring with total quotient ring T(R), and set such that for every and every Then is a ring homomorphism from T(R) into and restricted to R is also a ring homomorphism from R into given by for every At this setting, we introduce the notions of -piecewise Noetherian rings and -piecewise w-Noetherian rings and we show that the theories of -piecewise Noetherian (resp. -piecewise w-Noetherian) rings resemble those of piecewise Noetherian (resp. piecewise w-Noetherian) domains.

Let (S, *) be a ring with involution *. The involution is positive definite if, for all finite subsets {r,-} of S, £ w * = 0 implies all the r{ are zero. Then the set of self-adjoint elements of S, denoted S, possesses a natural partial ordering, with positive cone consisting of elements of the form Z l w * ; with this ordering, S is a directed partially ordered abelian group. Let Sb denote the set of bounded elements, that is, the set of elements s such that ss* is less than an integral multiple of 1 in this ordering. Then Sb is a *-subalgebra of S whenever S is an algebra over the rationals. We will be studying the objects Sb9 S , and (Sb) , from the point of view of their ordered structures. For instance, suppose S is a field, and * is the identity. Then S is formally real, and Sb must be Prufer domain, all of whose residue fields are themselves formally real (and in fact, are embeddable in the reals). Viewing Sb as a partially ordered abelian group with order unit 1 (indeed, Sb is the convex subgroup of S generated by 1), Sb has the Riesz decomposition property, and its normalized extremal states are precisely the ring homomorphisms into the reals. There is a natural mapping from the collection of total orderings of S to the set of extremal states of Sb9 and this in turn maps to Spec Sb (the prime ideal space of Sb); when S is even a real algebra much more can be said. If either S is a field and * is not the identity, or S is a quaternionic division algebra with the natural involution, essentially the same properties hold, with the appropriate modifications. A useful tool here is an involutory version of the Artin Schreier Theorem, about the existence of sufficiently many total orderings finer than the natural ordering. Studies are made of several specific bounded subrings. For instance, if S is the rational function field in one variable over the reals, then Sb is a Dedekind domain with class group of order 2, with spectrum the unit circle (in the point-open topology), and all of its maximal ideals are not principal. Expanding the scope of S somewhat, we next allow S to be a division

An element [Formula: see text] of a ring [Formula: see text] is called a quasi-idempotent if [Formula: see text] for some central unit [Formula: see text] of [Formula: see text], or equivalently, [Formula: see text], where [Formula: see text] is a central unit and [Formula: see text] is an idempotent of [Formula: see text]. A ring [Formula: see text] is called a quasi-Boolean ring if every element of [Formula: see text] is quasi-idempotent. A ring [Formula: see text] is called (strongly) quasi-clean if each of its elements is a sum of a quasi-idempotent and a unit (that commute). These rings are shown to be a natural generalization of the clean rings and strongly clean rings. An extensive study of (strongly) quasi-clean rings is conducted. The abundant examples of (strongly) quasi-clean rings state that the class of (strongly) quasi-clean rings is very larger than the class of (strongly) clean rings. We prove that an indecomposable commutative semilocal ring is quasi-clean if and only if it is local or [Formula: see text] has no image isomorphic to [Formula: see text]; For an indecomposable commutative semilocal ring [Formula: see text] with at least two maximal ideals, [Formula: see text]([Formula: see text]) is strongly quasi-clean if and only if [Formula: see text] is quasi-clean if and only if [Formula: see text], [Formula: see text] is a maximal ideal of [Formula: see text]. For a prime [Formula: see text] and a positive integer [Formula: see text], [Formula: see text] is strongly quasi-clean if and only if [Formula: see text]. Some open questions are also posed.

Let R R be a commutative ring with identity 1 ≠ 0 1 \ne 0 and let A A be a nonzero ideal of R R . A problem of current interest is to relate the notions of “projective ideal", “flat ideal” and “multiplication ideal". In this note we prove two results which show that the maximal ideals containing the annihilator of A A can play an important role in determining the relationship between these concepts. As a consequence we are able to prove that a finitely generated multiplication ideal in a semi-quasi-local ring is principal, that a finitely generated flat ideal having only a finite number of minimal prime divisors is projective and that for Noetherian rings or semihereditary rings, finitely generated multiplication ideals with zero annihilator are invertible.

In this paper, a new kind of graph on a commutative ring is introduced and investigated. Small intersection graph of a ring $R$, denoted by $G(R)$, is a graph with all non-small proper ideals of $R$ as vertices and two distinct vertices $I$ and $J$ are adjacent if and only if $I\cap J$ is not small in $R$. In this article, some interrelation between the graph theoretic properties of this graph and some algebraic properties of rings are studied. We investigated the basic properties of the small intersection graph as diameter, girth, clique number, cut vertex, planar property and independence number. Further, it is shown that the independence number of a small graph of a ring $R$ is equal to the number of its maximal ideals and the domination number of small graph is at most 2.

Let R R be a commutative Noetherian ring with unit. Let T T be the set of all elements of the total quotient ring of R R whose conductor to R R contains a power of a finite product of maximal ideals of R R . If A A is any ring such that R ⊂ A ⊂ T R \subset A \subset T , then A / x A A/xA is a finite R R module for any non-zero-divisor x x in R R . It follows that if, in addition, R R has no nonzero nilpotent elements, then any ring A A such that R ⊂ A ⊂ T R \subset A \subset T is Noetherian.

This paper discusses the concept of an n-valued group. The theory that is developed here can be regarded as having more precise conditions than those in the much more developed theory of hypergroups; it is also more oriented towards algebraic and topological ideas. A number of methods for constructing examples are given and the classification of the algebraic 2-valued group structures on the Riemann sphere is presented. We introduce the (purely algebraic) notion of an n-Hopf algebra and show that the ring of functions on an n-valued group has an n-Hopf algebra structure. The crucial property of an n-Hopf algebra is that the diagonal is an n-ring homomorphism that has a weaker multiplicative property than that satisfied by a ring homomorphism. This concept arises in a number of other contexts, in particular as a trace. Convolution of two ring homomorphisms from an n-Hopf algebra to any commutative ring yields an n-ring homomorphism. There is a duality between symmetric products of topological spaces and the set of all n-ring homomorphisms from the ring of functions on the space to C. The development of the properties of n-ring homomorphisms involves a number of combinatorial identities related closely to the classical properties of symmetric polynomials.

Let $R$ and $S$ be commutative rings with unity, $f\colon R\rightarrow S$ a ring homomorphism and $J$ an ideal of $S$. Then the subring $R\bowtie ^fJ:=\lbrace (a,f(a)+j)\mid a\in R$ and $j\in J\rbrace $ of $R\times S$ is called the amalgamation of $R$ with $S$ along $J$ with respect to $f$. In this paper, we determine when $R\bowtie ^fJ$ is a (generalized) filter ring.

The classes of <TEX>$G_C$</TEX> homological modules over commutative ring, where C is a semidualizing module, extend Holm and J<TEX>${\varnothing}$</TEX>gensen's notions of C-Gorenstein homological modules to the non-Noetherian setting and generalize the classical classes of homological modules and the classes of Gorenstein homological modules within this setting. On the other hand, transfer of homological properties along ring homomorphisms is already a classical field of study. Motivated by the ideas mentioned above, in this article we will investigate the transfer properties of C and <TEX>$G_C$</TEX> homological dimension.

Let R be a commutative unital ring and E a unital R-module. Then the canonical injective ring homomorphism from R into the idealization R(+) E is a minimal ring homomorphism if and only if E is a simple R-module. For E nonzero, R(+)E is not (R-algebra isomorphic to) an overring of R. If E 1 and E 2 are nonisomorphic simple R-modules, then R(+) E 1 and R(+) E 2 give minimal ring extensions of R which are not isomorphic as R-algebras. The ring of dual numbers over R is a minimal ring extension of R ⇔ R × R is a minimal ring extension of R ⇔ R is a field.

Here we discuss the role of the conductor of a ring extension vis-a-vis the descent of projectivity and injectivity. Regarding the former, the first result says that an injective homomorphism of commutative rings descends projectivity if it does so modulo the conductor. The cheap version—that with noetherian hypotheses—of the descent of projectivity by a finite homomorphism due to Gruson then follows easily. A carbon copy—with the natural modification—of the descent of injectivity is also proved. The statement of the results follows very closely the lines of similar work of Ferrand ([2]) on flat modules rather than those of the remarkable [3]. 1* Conductor and projectivity* Throughout rings will be commutative with identity element. The price to lift the restriction of commutativity would be to load the exposition with expressions like two-sided ideal, bi-module, etc., without any real gain, in view of the fact that for the applications the commutativity is critical. The main result of this section is, almost word for word, the projective analogue of [2]. Rather than using the results already obtained there, at no cost, we will provide complete proofs based on simple calculations. THEOREM 1.1. Let h: A —> B be an injective homomorphism of rings, I an ideal of A and E an A-module. Then E is A-projective if and only if B ®^ E is B-projective and E/IE is A/I-projectiv e in the following cases: (i) I is also a B-ideaL (ii) I is nilpotent.

A covariant functor has been defined in [1] which assigns to a commutative ring R an abelian monoid m(R). The elements of m(R) are equivalence classes (called homotopy classes) of homomorphisms of progenerator modules over R. The monoid homomorphism m(σ): m(R) → m(S) associated with a homomor-phism of rings σ: R → S is studied. Both necessary condi­tions and sufficient conditions for m(σ)1 {1} = {1}, for m(σ) to be injective and for m(σ) to be surjective are presented.