Rational points in henselian discrete valuation rings

Let R be a right hereditary domain in which all right ideals are two-sided (i.e., R is right invariant). We show that R is the intersection of generalized discrete valuation rings and that every right ideal is the product of prime ideals. This class of rings seems comparable with (and contains) the class of commutative Dedekind domains, but the rings considered here are in general not maximal orders and not Dedekind rings in the terminology of Robson [9]. The left order of a right ideal of such a ring is a ring of the same kind and the class contains right principal ideal domains in which the maximal right ideals are two-sided [6].

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Polynomial index in discrete valuation rings

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/012.2019.56.2.1424 ◽

2019 ◽

Vol 56 (2) ◽

pp. 260-266

Author(s):

Mohamed E. Charkani ◽

Abdulaziz Deajim

Keyword(s):

Lower Bound ◽

Valuation Ring ◽

Irreducible Polynomial ◽

Discrete Valuation Ring ◽

Field Extension ◽

Integral Closure ◽

Quotient Field ◽

Discrete Valuation Rings ◽

Valuation Rings ◽

Adic Valuation

Abstract Let R be a discrete valuation ring, its nonzero prime ideal, P ∈R[X] a monic irreducible polynomial, and K the quotient field of R. We give in this paper a lower bound for the -adic valuation of the index of P over R in terms of the degrees of the monic irreducible factors of the reduction of P modulo . By localization, the same result holds true over Dedekind rings. As an important immediate application, when the lower bound is greater than zero, we conclude that no root of P generates a power basis for the integral closure of R in the field extension of K defined by P.

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Direct sums of countably generated modules over complete discrete valuation rings

Proceedings of the American Mathematical Society ◽

10.1090/s0002-9939-1971-0276326-3 ◽

1971 ◽

Vol 28 (2) ◽

pp. 381-381 ◽

Cited By ~ 4

Author(s):

Chang Mo Bang

Keyword(s):

Direct Sums ◽

Discrete Valuation Rings ◽

Valuation Rings

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Integrable Derivations and Stable Equivalences of Morita Type

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091517000098 ◽

2018 ◽

Vol 61 (2) ◽

pp. 343-362 ◽

Cited By ~ 1

Author(s):

Markus Linckelmann

Keyword(s):

Hochschild Cohomology ◽

Block Theory ◽

Discrete Valuation Rings ◽

Symmetric Algebras ◽

Valuation Rings ◽

Transfer Maps ◽

Stable Equivalences ◽

Morita Type

AbstractUsing that integrable derivations of symmetric algebras can be interpreted in terms of Bockstein homomorphisms in Hochschild cohomology, we show that integrable derivations are invariant under the transfer maps in Hochschild cohomology of symmetric algebras induced by stable equivalences of Morita type. With applications in block theory in mind, we allow complete discrete valuation rings of unequal characteristic.

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EXPONENT MATRICES AND TILED ORDERS OVER DISCRETE VALUATION RINGS

International Journal of Algebra and Computation ◽

10.1142/s0218196705002724 ◽

2005 ◽

Vol 15 (05n06) ◽

pp. 997-1012 ◽

Cited By ~ 4

Author(s):

V. V. KIRICHENKO ◽

A. V. ZELENSKY ◽

V. N. ZHURAVLEV

Keyword(s):

Markov Chains ◽

Valuation Ring ◽

Discrete Valuation Ring ◽

Stochastic Matrices ◽

Doubly Stochastic ◽

Discrete Valuation Rings ◽

Valuation Rings ◽

Strongly Connected ◽

Exponent Matrix ◽

Finite Posets

Exponent matrices appear in the theory of tiled orders over a discrete valuation ring. Many properties of such an order and its quiver are fully determined by its exponent matrix. We prove that an arbitrary strongly connected simply laced quiver with a loop in every vertex is realized as the quiver of a reduced exponent matrix. The relations between exponent matrices and finite posets, Markov chains, and doubly stochastic matrices are discussed.

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