Arithmetical rings satisfy the radical formula
نویسندگان
چکیده
منابع مشابه
Almost clean rings and arithmetical rings
It is shown that a commutative Bézout ring R with compact minimal prime spectrum is an elementary divisor ring if and only if so is R/L for each minimal prime ideal L. This result is obtained by using the quotient space pSpec R of the prime spectrum of the ring R modulo the equivalence generated by the inclusion. When every prime ideal contains only one minimal prime, for instance if R is arith...
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In this paper we characterize the radical of an arbitrary submodule $N$ of a finitely generated free module $F$ over a commutatitve ring $R$ with identity. Also we study submodules of $F$ which satisfy the radical formula. Finally we derive necessary and sufficient conditions for $R$ to be a Pr$ddot{mbox{u}}$fer domain, in terms of the radical of a cyclic submodule in $Rbigopl...
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It is proved that localizations of injective R-modules of finite Goldie dimension are injective if R is an arithmetical ring satisfying the following condition: for every maximal ideal P , RP is either coherent or not semicoherent. If, in addition, each finitely generated R-module has finite Goldie dimension, then localizations of finitely injective R-modules are finitely injective too. Moreove...
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In this paper we consider several constructions which from a given B-product ∗B lead to another one ∗̃B . We shall be interested in finding what algebraic properties of the ring RB = 〈CN, +, ∗B 〉 are shared also by the ring RB̃ = 〈C N, +, ∗B 〉. In particular, for some constructions the rings RB and RB̃ will be isomorphic and therefore have the same algebraic properties. §
متن کاملRadical Extensions of Rings
Jacobson's generalization [5, Theorem 8] of Wedderburn's theorem [8] states that an algebraic division algebra over a finite field is commutative. These algebras have the property that some power2 of each element lies in the center. Kaplansky observed in [7] that any division ring, or, more generally, any semisimple ring, in which some power of each element lies in the center is commutative. Ka...
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ژورنال
عنوان ژورنال: Journal of Commutative Algebra
سال: 2012
ISSN: 1939-2346
DOI: 10.1216/jca-2012-4-2-293