Prime Ideals and Integral Dependence
نویسنده
چکیده
Let 9t and © be commutative rings such that © contains, and has the same identity element as, 9Î. If p and $ are prime ideals in SK and © respectively such that ^P\9t = p then we shall say that $ lies over, or contracts to, p. If over every prime ideal in dt there lies a prime ideal in ©, we shall say that the "lying-over" theorem holds for the pair of rings 9Î and ©. Suppose now that q and p are prime ideals in 91 such that qCp. If for every prime ideal O in © lying over q there exists a prime ideal $ in © lying over p and containing O, then the "going-up" theorem will be said to hold for 9t and ©. Similarly, if for every prime ideal $ in © lying over p there exists a prime ideal O in © lying over q and contained in ty, then the "going-down " theorem will be said to hold. Below we are concerned with the case where © is integrally dependent on 9î. In this case we shall prove the "lying-over" and "going-up" theorems (§1). With certain additional conditions on 9Î and ©, also the "going-down theorem is proved (§2). Counterexamples are given to show that none of these conditions can be omitted (§3). All of the results of this paper (except Theorem 7) have been proved by Krull when the rings are free from zero-divisors. The present proofs are essentially simpler than Krull's and at the same time do not require that the rings be integral domains.
منابع مشابه
AN INTEGRAL DEPENDENCE IN MODULES OVER COMMUTATIVE RINGS
In this paper, we give a generalization of the integral dependence from rings to modules. We study the stability of the integral closure with respect to various module theoretic constructions. Moreover, we introduce the notion of integral extension of a module and prove the Lying over, Going up and Going down theorems for modules.
متن کاملTitle of thesis: CLASSIFICATION OF PRIME IDEALS IN INTEGRAL GROUP ALGEBRAS OF FINITE ABELIAN GROUPS
Title of thesis: CLASSIFICATION OF PRIME IDEALS IN INTEGRAL GROUP ALGEBRAS OF FINITE ABELIAN GROUPS Heather Mallie McDonough, Master of Arts, 2005 Thesis directed by: Professor William Adams Department of Mathematics Let Z[G] be the integral group algebra of the group G. In this thesis, we consider the problem of determining all prime ideals of Z[G] where G is both finite and abelian. Because o...
متن کاملGraded Prime Ideals Attached to a Group Graded Module
Let $G$ be a finitely generated abelian group and $M$ be a $G$-graded $A$-module. In general, $G$-associated prime ideals to $M$ may not exist. In this paper, we introduce the concept of $G$-attached prime ideals to $M$ as a generalization of $G$-associated prime ideals which gives a connection between certain $G$-prime ideals and $G$-graded modules over a (not necessarily $G$-graded Noetherian...
متن کاملI-prime ideals
In this paper, we introduce a new generalization of weakly prime ideals called $I$-prime. Suppose $R$ is a commutative ring with identity and $I$ a fixed ideal of $R$. A proper ideal $P$ of $R$ is $I$-prime if for $a, b in R$ with $ab in P-IP$ implies either $a in P$ or $b in P$. We give some characterizations of $I$-prime ideals and study some of its properties. Moreover, we give conditions ...
متن کاملOn generalisations of almost prime and weakly prime ideals
Let $R$ be a commutative ring with identity. A proper ideal $P$ of $R$ is a $(n-1,n)$-$Phi_m$-prime ($(n-1,n)$-weakly prime) ideal if $a_1,ldots,a_nin R$, $a_1cdots a_nin Pbackslash P^m$ ($a_1cdots a_nin Pbackslash {0}$) implies $a_1cdots a_{i-1}a_{i+1}cdots a_nin P$, for some $iin{1,ldots,n}$; ($m,ngeq 2$). In this paper several results concerning $(n-1,n)$-$Phi_m$-prime and $(n-1,n)$-...
متن کامل