Projective Modules Over Central Separable Algebras

In (2), M. Auslander and O. Goldman laid the foundations for the study of central separable algebras. For unexplained terminology and notation, see (2). Here we are interested in projective modules and the ideal structure of a central separable algebra A over some special commutative rings K. When K is a field, one consequence of Wedderburn's Theorem is that there is a unique (up to isomorphism) irreducible A-module. We show here that if K is a commutative ring with a finite number of maximal ideals (semi-local) and with no idempotents other than 0 and 1 or if K is the ring of polynomials in one variable over a perfect field, then there is a unique (up to isomorphism) indecomposable finitely generated projective A-module. An example in (3) shows that this result fails if one only assumes that K is a principal ideal domain.

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On a Class of Projective Modules Over Central Separable Algebras

Canadian Mathematical Bulletin ◽

10.4153/cmb-1971-072-6 ◽

1971 ◽

Vol 14 (3) ◽

pp. 415-417 ◽

Cited By ~ 2

Author(s):

George Szeto

Keyword(s):

Finite Number ◽

Commutative Ring ◽

Projective Module ◽

Projective Modules ◽

Finitely Generated ◽

Perfect Field ◽

Maximal Ideals ◽

Ring Of Polynomials ◽

Separable Algebras

In [5], DeMeyer extended one consequence of Wedderburn's theorem; that is, if R is a commutative ring with a finite number of maximal ideals (semi-local) and with no idempotents except 0 and 1 or if R is the ring of polynomials in one variable over a perfect field, then there is a unique (up to isomorphism) indecomposable finitely generated projective module over a central separable R-algebra A.

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A characterization of modules embedding in products of primes and enveloping condition for their class

Journal of Algebra and Its Applications ◽

10.1142/s0219498815500516 ◽

2015 ◽

Vol 14 (04) ◽

pp. 1550051 ◽

Cited By ~ 1

Author(s):

N. Dehghani ◽

M. R. Vedadi

Keyword(s):

Finite Number ◽

Commutative Rings ◽

Dedekind Domain ◽

Noetherian Rings ◽

Artinian Rings ◽

Morita Equivalent ◽

Maximal Ideals ◽

Weakly Compressible ◽

Strongly Regular

Modules (cogenerated by nonzero their submodules) having nonzero square homomorphism in nonzero submodules are said (prime) weakly compressible (wc). Such modules are semiprime (i.e. they are cogenerated by their essential submodules). For many rings R, including commutative rings, it is proved that wc modules are isomorphic submodules of products of prime modules, and the converse holds precisely when the class of wc modules is enveloping for mod-R or equivalently, every semiprime module is wc. Semi-Artinian rings R have the latter property and the converse is true when R is strongly regular. Duo Noetherian rings over which wc modules form an enveloping class are shown to have a finite number maximal ideals. If R is Morita equivalent to a Dedekind domain, then the class of wc R-modules is enveloping if and only if R is simple Artinian or J (R) ≠ 0.

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Ideal structure of rings of analytic functions with non-Archimedean metrics

Journal of Algebra and Its Applications ◽

10.1142/s0219498823500111 ◽

2021 ◽

Author(s):

Nicholas Bruno

Keyword(s):

Analytic Functions ◽

Complex Analysis ◽

Entire Functions ◽

Prime Ideals ◽

Ideal Structure ◽

Finitely Generated ◽

Finite Radius ◽

Maximal Ideals ◽

Divisibility Properties ◽

The Ideal

The work of Helmer [Divisibility properties of integral functions, Duke Math. J. 6(2) (1940) 345–356] applied algebraic methods to the field of complex analysis when he proved the ring of entire functions on the complex plane is a Bezout domain (i.e. all finitely generated ideals are principal). This inspired the work of Henriksen [On the ideal structure of the ring of entire functions, Pacific J. Math. 2(2) (1952) 179–184. On the prime ideals of the ring of entire functions, Pacific J. Math. 3(4) (1953) 711–720] who proved a correspondence between the maximal ideals within the ring of entire functions and ultrafilters on sets of zeroes as well as a correspondence between the prime ideals and growth rates on the multiplicities of zeroes. We prove analogous results on rings of analytic functions in the non-Archimedean context: all finitely generated ideals in the ring of analytic functions on an annulus of a characteristic zero non-Archimedean field are two-generated but not guaranteed to be principal. We also prove the maximal and prime ideal structure in the non-Archimedean context is similar to that of the ordinary complex numbers; however, the methodology has to be significantly altered to account for the failure of Weierstrass factorization on balls of finite radius in fields which are not spherically complete, which was proven by Lazard [Les zeros d’une function analytique d’une variable sur un corps value complet, Publ. Math. l’IHES 14(1) (1942) 47–75].

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THE IDEALS OF AN IDEAL EXTENSION

Journal of Algebra and Its Applications ◽

10.1142/s0219498810003999 ◽

2010 ◽

Vol 09 (03) ◽

pp. 407-431 ◽

Cited By ~ 4

Author(s):

ZACHARY MESYAN

Keyword(s):

Ideal Extension ◽

Ideal Structure ◽

Arbitrary Ring ◽

Maximal Ideals ◽

Associative Rings ◽

The Ideal

Given two unital associative rings R ⊆ S, the ring S is said to be an ideal (or Dorroh) extension of R if S = R ⊕ I, for some ideal I ⊆ S. In this note, we investigate the ideal structure of an arbitrary ideal extension of an arbitrary ring R. In particular, we describe the Jacobson and upper nil radicals of such a ring, in terms of the Jacobson and upper nil radicals of R, and we determine when such a ring is prime and when it is semiprime. We also classify all the prime and maximal ideals of an ideal extension S of R, under certain assumptions on the ideal I. These are generalizations of earlier results in the literature.

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Criteria and Methods for Reliable Three-Dimensional Image Reconstruction

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100051669 ◽

1974 ◽

Vol 32 ◽

pp. 330-331

Author(s):

R. A. Crowther

Keyword(s):

Image Reconstruction ◽

Finite Number ◽

Density Distribution ◽

Electron Micrographs ◽

Mathematical Problem ◽

Three Dimensional ◽

Ideal Case ◽

Dimensional Image ◽

General Object ◽

The Ideal

The reconstruction of a three-dimensional image of a specimen from a set of electron micrographs reduces, under certain assumptions about the imaging process in the microscope, to the mathematical problem of reconstructing a density distribution from a set of its plane projections.In the absence of noise we can formulate a purely geometrical criterion, which, for a general object, fixes the resolution attainable from a given finite number of views in terms of the size of the object. For simplicity we take the ideal case of projections collected by a series of m equally spaced tilts about a single axis.

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The ideal structure of XX

Semigroup Forum ◽

10.1007/bf02573436 ◽

1984 ◽

Vol 30 (1) ◽

pp. 41-51 ◽

Cited By ~ 4

Author(s):

Neil Hindman ◽

Paul Milnes

Keyword(s):

Ideal Structure ◽

The Ideal

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Falling -Ideals in -Algebras

Discrete Dynamics in Nature and Society ◽

10.1155/2011/516418 ◽

2011 ◽

Vol 2011 ◽

pp. 1-14 ◽

Cited By ~ 2

Author(s):

Young Bae Jun ◽

Sun Shin Ahn ◽

Kyoung Ja Lee

Keyword(s):

Theoretical Approach ◽

Ideal Structure ◽

Fundamental Properties ◽

The Ideal

Based on the theory of a falling shadow which was first formulated by Wang (1985), a theoretical approach of the ideal structure in -algebras is established. The notions of a falling -subalgebra, a falling -ideal, a falling -ideal, and a falling -ideal of a -algebra are introduced. Some fundamental properties are investigated. Relations among a falling -subalgebra, a falling -ideal, a falling -ideal, and a falling -ideal are stated. Characterizations of falling -ideals and falling -ideals are discussed. A relation between a fuzzy -subalgebra and a falling -subalgebra is provided.

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