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Im Jahr 2003 fand in der Göttinger Jakobikirche eine hochkarätige und deshalb zu Recht viel beachtete Vorlesungsreihe zur Wirkungsgeschichte der Bibel statt. Namhafte Vertreter bester Göttinger Philologie fanden hier ihr Publikum - interessiert, oft gelehrt, aber in seinem Interesse nicht auf eine fachwissenschaftliche Binnendiskussion beschränkt. Diese nun gedruckt vorliegende Vorlesungsreihe ist nicht nur ein Beispiel für das gute Zusammenleben von Universität und Stadt, wie es typisch für Göttingen ist. Sie ist darüberhinaus Beispiel für ein fruchtbares Gespräch zwischen Wissenschaft und Kirche, wie es auch außerhalb der akademischen Theologie möglich und nötig ist. A top-flight and with justification noticed series of lectures on the subject of the bible´s effect had been held at the Göttinger Jacobikirche in 2003. It is an example for the tipical living-together of university and city and for a productive dialogue between sciences and church, which is even outside academic theology possible and needed. peerReviewed

In this paper R is a commutative noetherian local ring with unit element 1 and M is its maximal ideal. Let K be the residue field R/M and let {t1,t2,…, tn) be a minimal system of generators for M. By a complex R1. . ., Tp> we mean an R-algebra* obtained by the adjunction of the variables T1. . ., Tp of degree 1 which kill t1,…, tp. The main purpose of this paper is, among other things, to construct an R-algebra resolution of the field K, so that we can investigate the relationship between the homology algebra H (R < T1,…, Tn>) and the homological invariants of R such as the algebra TorR(K, K) and the Betti numbers Bp = dimk TorR(K, K) of the local ring R. The relationship was initially studied by Serre [5].

Let R be a ring with identity element 1, and let S be a subring of R containing 1. We consider R-modules on which 1 acts as the identity map, and we shall simultaneously regard such R-modules as S-modules in the natural way. In [4], we have defined the relative analogues of the functors of Cartan-Eilenberg [1], and we have briefly treated the corresponding relative analogues of module dimension and global ring dimension.

In this note we shall prove some supplementary results of Jacobson [5] concerning Kronecker products of primitive algebras and those of P.M.I. algebras (that is, algebras with faithful minimal one sided ideals) and study their applications....

LetΩbe an algebraic number field of finite degree, which we fix once for all, and letKbe a cyclic extension overΩsuch that the degree ofK/Ωis a powerof a prime numberl. It is obvious that the norm groupNK/ΩeKof the unit group ekofK, being a subgroup of the unit group e of Ω contains the groupconsisting of all-th powersof ε∈e.

Introduction In this discussion every module over a ring R will be understood to be a left i2-module. R will always have a unit, and every module will be unitary. The aim of this paper is to study the structure and properties of injective modules, particularly over Noetherian rings. B. Eckmann and A. Schopf have shown that if M is a module over any ring, then there exists a unique, minimal, injective module E(M) containing it. The module E(M) will be a major tool in our investigations, and we shall systematically exploit its properties. In § 1 we show that if a module M has a maximal, injective submodule C (as is the case for left-Noetherian rings), then C contains a carbon-copy of every injective submodule of M, and MjC has no injective submodules different from 0. Although C is unique up to an automorphism of My C does not in general contain every injective submodule of M, In fact, the sum of two injective submodules of a module is always injective if and only if the ring is left-heredita ry. In § 2 we show that for any ring R a module E is an indecomposable, injective module if and only if E = E(R\J)y where J is an irreducible, left ideal of R. We prove that if R is a left-Noetherian ring, then every injective Jϋ-module has a decomposition as a direct sum of indecomposable, injective submodules. Strong uniqueness assertions can be made concerning such decompositions over any ring. In § 3 we take R to be a commutative, Noetherian ring, and P to be a prime ideal of R. We prove there is a one-to-one correspondence between the prime ideals of R and the indecomposable, injective Rmodules given by P**E(RjP). We examine the structure of the module E = E{RjP)y and show that if At is the annihilator in E of P\ then E = U At and -4ί+1/At is a finite dimensional vector space over the quotient field of R/P. The ring of iϋ-endomorphi sms of E is isomorphic in a natural way to Rp, the completion of the ring of quotients of R with respect to R-P. As an ^-module E is an injective envelope of RpjP, where P is the maximal ideal of Rp. If P is a maximal ideal of Ry then E is a countably generated β-module. Every indecomposable, injective i2-module is finitely generated if and only if R has the minimum condition on ideals. In § 4 we take R to be a commutative, Noetherian, complete, local ring, P the maximal ideal of R and E = E{RjP). Then the eontravariant

Let R be a Dedekind domain and K the quotient field of R. Let Σ be a central simple K-algebra of finite dimension over K....

Let R be a Dedekind domain and K its quotient field, and Σ a central simple K-algebra with finite rank over K. We have already defined a hereditary order Λ over R in Σ in [5], [6] and [7], namely Λ is a hereditary ring in Σ such that Λ is a finitely generated R-module and ΛK=Σ....

In this paper we study two features of relative homological algebra of Frobenius extensions, the one is the relative dimension and the other is the relative complete resolution. The theory of Frobenius extensions was developed by F. Kasch [4] as a generalization of Frobenius algebras. We deal with only Frobenius extensions of finite rank (see §2 for the definition).

Introduction. Let S be the integral closure of a complete discrete rank one valuation ring R in a finite Galois extension of the quotient field of R, and let G denote the Galois group of the quotient field extension. Auslander and Rim have shown in [3] that the trivial crossed product Δ (1, S, G) is an hereditary order if and only if 5 is a tamely ramified extension of R. And the author has proved in [7] that if the extension S of R is tamely ramified then the crossed product Δ(f, 5, G) is a Π-principal hereditary order for each 2-cocycle f in Z2(G, U(S)). (See Section 1 for the definition of Π-principal hereditary order.) However, the author has exhibited in [8] an example of a crossed product Δ(f, S, G) which is a Π-principal hereditary order in the case when S is a wildly ramified extension of R.

The questions concerning the dimension of the tensor product of two K-algebras have turned out to be surprisingly difficult. In this paper we follow a method using spectral sequences (§§1-3) which in some concrete cases yields complete results (§§4-5). In particular, complete results are obtained when r is a ring of matrices, triangular matrices, polynomials or rational functions, so that in the first three cases is respectively the ring of matrices, triangular matrices or polynomials with coefficients in the arbitrary algebra A.

Let R be a Dedekind domain and Σ a central simple K-algebra, where K is the quotient field of R. M. Auslander and O. Goldman have considered maximal orders in Σ from point of view of homological method in [1] and introduced the notion of hereditary orders in Σ....

In § 1 of this note we first define the trace of an endomorphism of a projective module P over a non-commutative ring A. Then we call the trace of the identity the rank element r(P) of P, which we shall illustrate by several examples. For a projective module P over the groupalgebra of a finite group G, the rank element of P is essentially the character of G in P. In § 2 we prove that under certain assumption two projective modules Pi and P2 over an algebra over a complete local ring o are isomorphic if and only if their rank elements are identical. This is a type of proposition asserting that two representations are equivalent if and only if their characters are identical, and in fact, when A is the groupalgebra, the above theorem may be considered as another formulation of Swan’s local theorem [9]).

Let there be given a compact topological manifold Mn. If Mn admits a “triangulation” it is known that the fundamental invariants, namely the connectivities of Mn over fields, the Betti numbers and torsion coefficients over Z of the singular homology groups of Mn, are finite and calculable. However it is not known that a “triangulation” of Mn always exists when n > 3.

The motivation for the results in this note comes from a theorem of Macaulay. Let f 1, …, fn be elements of a polynomial ring R over a field, and let I be the ideal they generate. Assume I R and rank (I) = n. Then the theorem of Lasker and Macaulay asserts that I is unmixed (all prime ideals belonging to I have rank n). Macaulay [1, p. 51] proved further that any power of I is unmixed.

In this note we shall give a modified definition of cohomology groups for algebras. For a class of (infinite rank) algebras (which includes Frobenius algebras, group rings of infinite groups and division algebras of countable rank over fields), these groups can be characterized in a manner similar to the cohomology groups in the * Almost Zero theory’ of B. Eckmann [2]. Actually, in the case of group rings, these coincide with the cohomology groups in the almost zero theory.

When we consider modules A over a ring R which is not a commutative integral domain, the usual torsion theory becomes somewhat inadequate, since zero-divisors of R are disregarded and since the torsion elements of A do not in general form a submodule. In this paper we shall try to remedy such defects by modifying the fundamental notions such as torsion modules, divisible modules, etc.

Let J be a directed set and let {Λj, φij} be a direct system of rings indexed by J and with limit Λ. Let {Aj, φij} be a direct system of groups indexed by J. Assume that each Aj is a left Λj-module and that φij(λa) = φij(λ) φij(a) for each λ ∈ Λj, a ∈ Aj. Then the limit A of (Aj, φij) is a left Λ-module.

In this note we shall define the weak dimension of algebras Λ, analogous to the dimension of algebras in Cartan and Eilenberg [6], Ch. IX. In section 1 we shall characterize the algebras with the weak dimension zero, and study some properties of the weak dimension of the tensor product of two algebras, and we shall completely determine the weak dimension of fields....

Let G be a group and H a subgroup of G. With a left G-module M, relative cohomology groups Hn(G, H, M) of G on M, relative to H, have been defined by Adamson [1] and may be expressed as (Z, M) in the notation of relative homological algebra of Hochschild [2], where Z denotes the G-module of rational integers (acted by G trivially). Regarding M as a right G-module, (M, Z) are similarly relative homology groups Hn(G, H, M). In case H is of finite index in G, Hochschild [2] defines further negative-dimensional relative homology and cohomology groups.

The purpose of this paper is to find out what can be learned about valuation rings, and more generally Prufer rings, from a study of their injective modules. The concept of an almost maximal valuation ring can be reformulated as a valuation ring such that the images of its quotient field are injective. The integral domains with this latter property are found to be the Prufer rings with a (possibly) weakened form of linear precompactness for their quotient fields.