Commutative Artinian Principal Ideal Rings

McLean, K. Robin

doi:10.1112/plms/s3-26.2.249

Cited by 26 publications

(31 citation statements)

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“…Moreover, in [5,Theorems 3.5,3.6] the author classifies all tamely ramified (i.e., p m in the aforementioned theorem) artinian local PIRs up to isomorphism by taking advantage of the established results on v-rings.…”

Section: Introductionmentioning

confidence: 98%

“…83.) that if P is a field of characteristic p, then there exists an unramified complete discrete valuation ring, V , of characteristic 0 whose residue field is isomorphic to P. This ring V is uniquely determined up to isomorphism by P and is called the v-ring with residue field P. In [5,Theorem 3.2], the author uses the 4728 WU ET AL.…”

Section: Introductionmentioning

confidence: 98%

“…For a ring R, let U R be the set of invertible elements of R, J R the Jacobson radical of R and char R the characteristic of R. A ring R is called a principal ideal ring (PIR) if each ideal of R is a principal ideal. For a local ring R with a finite number of ideals (i.e., an artinian local ring R), R is a PIR if and only if the maximal ideal is cyclically generated ([1, Proposition 8.8] or [5,Theorem 2.1]). Furthermore, the number of nontrivial ideals of R is s if s + 1 is the nilpotency index of .…”

Section: Introductionmentioning

confidence: 99%

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The Structure of Finite Local Principal Ideal Rings

2012

Communications in Algebra

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Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

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The Structure of Finite Local Principal Ideal Rings

2012

Communications in Algebra

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“…Any proper image of a PID is a commutative Artinian ring, hence the direct sum of local Artinian rings [15]. Since each of these local rings is the image of a PID, its maximal ideal is principal, i.e.…”

Section: Example If D Is a Division Ring And Rmentioning

confidence: 99%

“…If R is a principal ideal domain and P is a prime ideal of R, then R/(P") is an LRFCR. The commutative LRFCRs are precisely the proper homomorphic images of complete discrete valuation rings [15].…”

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confidence: 99%

Deductive Varieties of Modules and Universal Algebras

Hogben¹,

Bergman²

1985

Transactions of the American Mathematical Society

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Abstract.A variety of universal algebras is called deductive if every subquasivariety is a variety. The following results are obtained: (1) The variety of modules of an Artinian ring is deductive if and only if the ring is the direct sum of matrix rings over local rings, in which the maximal ideal is principal as a left and right ideal. (2) A directly representable variety of finite type is deductive if and only if either (i) it is equationally complete, or (ii) every algebra has an idempotent element, and a ring constructed from the variety is of the form (1) above. This paper initiates a study of varieties of universal algebras with the property that every subquasivariety is a variety. We call such varieties deductive. In §2 it is proved that the variety of left modules of a left Artinian ring R is deductive if and only if R is the direct sum of matrix rings over local rings in which the maximal ideal is principal as a left and right ideal. In §3 we consider varieties of universal algebras of finite type that are directly representable. A variety is directly representable if it is generated by a finite algebra and has, up to isomorphism, only finitely many directly indecomposable finite algebras. Such a variety is deductive if and only if either (i) it is equationally complete, or (ii) every algebra has an idempotent element and a certain ring, constructed from the variety, has the properties listed above.Deductive varieties were first considered by those studying the varieties arising in algebraic logic. (The Russian equivalent for "deductive" would be "hereditarily structurally complete, in the finitary sense.") Roughly speaking, subvarieties correspond to new axioms added to the logic under consideration, while subquasivarieties correspond to new rules of inference. Thus, in the logic associated with a deductive variety, every rule of inference can be replaced by a set of logical axioms. Some other results on deductive varieties can be found in Tsitkin [19] on varieties of Hey ting algebras, and in Igosin [10] on some deductive varieties of lattices. This paper got its start with the consideration of discriminator varieties, which falls under the heading of algebraic logic. From there the material in §3 developed, reducing the problem of characterizing those directly representable varieties that are deductive to the corresponding problem for finite rings.

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A Topological Characterization of Hjelmslev’s Classical Geometries

Lorimer

1985

Rings and Geometry

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Commutative Artinian Principal Ideal Rings

Cited by 26 publications

References 5 publications

The Structure of Finite Local Principal Ideal Rings

The Structure of Finite Local Principal Ideal Rings

Deductive Varieties of Modules and Universal Algebras

A Topological Characterization of Hjelmslev’s Classical Geometries

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