A proper ideal I of a ring R is said to be strongly irreducible if for each pair of ideals A and B of R, implies that either or . In this paper we study strongly irreducible ideals in different rings. The relations between strongly irreducible ideals of a ring and strongly irreducible ideals of localizations of the ring are also studied. Furthermore, a topology similar to the Zariski topology related to strongly irreducible ideals is introduced. This topology has the Zariski topology defined by prime ideals as one of its subspace topologies.

Heitmann’s proof of the direct summand conjecture has opened a new approach to the study of homological conjectures in mixed characteristic. Inspired by his work and by the methods of almost ring theory, we discuss a normalized length for certain torsion modules, which was introduced by Faltings. Using the normalized length and the Frobenius map, we prove some results of local cohomology for local rings in mixed characteristic, which has an immediate implication for the subject of splinters studied by Singh.

Let M and N be finitely generated and graded modules over a standard positive graded commutative Noetherian ring R, with irrelevant ideal R+. Let be the nth component of the graded generalized local cohomology module . In this paper we study the asymptotic behavior of Assf R+ () as n → –∞ whenever k is the least integer j for which the ordinary local cohomology module is not finitely generated.

For any abelian group G and any function f: G → G we define a commutative binary operation or ‘multiplication’ on G in terms of f. We give necessary and sufficient conditions on f for G to extend to a commutative ring with the new multiplication. In the case where G is an elementary abelian p–group of odd order, we classify those functions which extend G to a ring and show, under an equivalence relation we call weak isomorphism, that there are precisely six distinct classes of rings constructed using this method with additive group the elementary abelian p–group of odd order p2.

In this paper we begin with a short, direct proof that the Banach algebra B(l1) is not amenable. We continue by showing that various direct sums of matrix algebras are not amenable either, for example the direct sum of the finite dimensional algebras is no amenable for 1 ≤ p ≤ ∞, p ≠ 2. Our method of proof naturally involves free group algebras, (by which we mean certain subalgebras of B(X) for some space X with symmetric basis—not necessarily X = l2) and we introduce the notion of ‘relative amenability’ of these algebras.

This work is concerned with the question of when a von Neumann regular ring is expressible as a directed union of Artinian subrings.

Let R be an integral domain with quotient field K and let X be an indeterminate. A result of W. C. Waterhouse states that, if each quadratic polynomial f ∈ R[X] which factors into linear polynomials in K[X] also factors into linear polynomials in R[X], then every irreducible element in R is prime. In this note the rings which satisfy the hypothesis of this theorem are characterized, and compared to the rings for which each polynomial f ∈ R[X] which factors into two polynomials of positive degree in K[X] also factors into two polynomials of positive degree in R[X]. Relevant examples are furnished via the pullback construction.

Let R be a commutative Noetherian ring with nonzero identity and let M be a finitely generated R-module. In this paper, we prove that if an ideal I of R is generated by a u.s.d-sequence on M then the local cohomology module (M) is I-cofinite. Furthermore, for any system of ideals Φ of R, we study the cofiniteness problem in the context of general local cohomology modules.

Let M be a commutative cancellative atomic monoid. We consider the behaviour of the asymptotic length functions and on M. If M is finitely generated and reduced, then we present an algorithm for the computation of both and where x is a nonidentity element of M. We also explore the values that the functions and can attain when M is a Krull monoid with torsion divisor class group, and extend a well-known result of Zaks and Skula by showing how these values can be used to characterize when M is half-factorial.

Let R be a Noetherian local ring with maximal ideal m and lull ring of fractions Q. In this paper we consider a numerical function EHI: ℤ → ℤ, where I is an m-primary ideal of R, that coincides with the Hilbert function HI for positive values and that takes account of the fractional powers of I for negative values. We focus our attention on the one-dimensional case. Among other results we characterize one-dimensional Gorenstein local rings by means of the symmetry of EHR in Theorem 2.1, we show that the extended Hilbert function is not determined by the Hilbert function in Example 2.2. and we generalize to m-primary ideals the upper bound for e1(m) given by Matlis for the maximal ideal.

Given polynomials a and b over an integral domain R, their tensor product (denoted a ⊗ b) is a polynomial over R of degree deg(a) deg(b) whose roots comprise all products αβ, where α is a root of a, and β is a root of b. This paper considers basic properties of ⊗ including how to factor a ⊗ b into irreducibles factors, and the direct sum decomposition of the ⊗-product of fields.

It is shown that an integral domain R has the property that every pure submodule of a finite direct sum of ideals of R is a summand if and only if R is an h-local Prüfer domain; equivalently, (J + K:I) = (J:I) + (K:I) for all ideals I, J and K of R. These results are extended to submodules of the quotient field of an integral domain.

For i = 1,…, n let ai be a homogeneous polynomial of degree ri(>0) in the graded polynomial ring R[x1, …, xm], or R[x] for short, where R is a commutative ring with unity and x1, …, xm are indeterminates of degree 1. Let of degree - 1 be a formal inverse of xj and let U denote the graded R[x]-module In [2, §2] we introduced a graded complex of r-modules.

In what follows, R will denote a commutative domain with 1, and Q(≠R) its field of quotients, which is viewed here as an R-module. By RP we denote the localization of R at the maximal ideal P, and more generally, by MP = Rp⊗RM the localization of the R-module M at P, which we define to be the P-component of M. The symbol R* will mean the multiplicative monoid of nonzero elements of R. For a submonoid S of R*, Rs will denote the localization of R at S.

Suppose D is an integral domain with quotient field K and that L is an extension field of K. We show in Theorem 4 that if the complete integral closure of D is an intersection of Archimedean valuation domains on K, then the complete integral closure of D in L is an intersection of Archimedean valuation domains on L; this answers a question raised by Gilmer and Heinzer in 1965.

Let ℜ be the class of commutative rings R with comparable regular elements, that is, given two non zero-divisors in R, one divides the other. Applying the notion of V-valuation due to Harrison and Vitulli, we define the class V-val of V-valuated rings, which is contained in ℜ and contains the class of Manis valuation rings. We prove that these inclusions of classes are both proper. We investigate Prüfer rings inside ℜ, showing that there exist Prüfer rings which lie in ℜ but not in V-val; we prove that a ring R is a Prüfer valuation ring if and only if it is Prüfer and V-valuated, if and only if its lattice of regular ideals is a chain. Finally, we introduce and investigate the ideal I∞ of a ring R ∈ ℜ, which corresponds to the counterimage of ∞, whenever R is V-valuated.

Let A be a subring of a commutative ring B. If the natural mapping from the prime spectrum of B to the prime spectrum of A is injective (respectively bijective) then the pair (A, B) is said to have the injective (respectively bijective) Spec-map. We give necessary and sufficient conditions for a pair of rings A and B graded by a free abelian group to have the injective (respectively bijective) Spec-map. For this we first deal with the polynomial case. Let l be a field and k a subfield. Then the pair of polynomial rings (k[X], l[X]) has the injective Spec-map if and only if l is a purely inseparable extension of k.

Let D be an atomic integral domain (i.e., a domain in which each nonzero nonunit of D can be written as a product of irreducible elements) and k any positive integer. D is known as a half factorial domain (HFD) if for any irreducible elements α1, …, αn, β1, …, βm of D the equality α1… αn = β1… βm implies that n = m. In [5] the present authors define D to be a k-half factorial domain (k-HFD) if the equality above along with the fact that n or m ≤ k implies that n = m. In this paper we consider the k-HFD property in Dedekind domains with small class group and prove the following Theorem: if D is a Dedekind domain with class group of order less than 16 then D is k-HFD for some integer k > 1, if, and only if, D is HFD.

Let R be a commutative, Noetherian ring and let Q be the total quotient ring of R. We shall call B an intermediate ring if R ⊂ B ⊂ Q. In [S] it is proved, for an integral domain R, that if R ⊂ B ⊂ Rf where B is flat over R, then B is a finitely generated R-algebra. We observe that the result holds for any commutative, Noetherian ring where f is a non-zero divisor. Our proof [Theorem 1.1] is a little different and straight; it is given for completeness. The idea of the proof in [S] lies in finding an ideal I of R such that IB = B, and for any λ∈I, b∈B there exists m ≥ 1 such that λmb ∈ R. We shall show that even if an intermediate ring B is finitely generated R-algebra, there may not exist any ideal I of R such that IB = B, moreover, if B is not finitely generated R-algebra, we may have IB = B for some ideal I in R.

A lattice-ordered power series algebra of a totally ordered field over a rooted abelian group may be constructed in a way that is arbitrary only in requiring that a factor set be chosen in the field and an extended total order be chosen on the group modulo its torsion subgroup. The resulting algebra is a field if and only if the subalgebra of elements with torsion support form a field. It follows that if the torsion subgroup may be independently embedded in the algebraic closure of the totally ordered field, or if the resulting algebra has no zero-divisors, then the algebra is a field. The set of supporting subsets for the power series may be characterized abstractly in such a way that previous representation theorems of lattice-ordered fields into power series algebras may be applied to produce representations into power series fields.