منابع مشابه
Commutative Regular Rings without Prime Model Extensions
It is known that the theory K of commutative regular rings with identity has a model completion K . We show that there exists a countable model of K which has no prime extension to a model of K'. If K and K ate theories in a first order language L, then K is said to be a model completion of K if K extends K, every model of K can be embedded in a model of K , and for any model A of K and models ...
متن کاملOn Some Properties of Extensions of Commutative Unital Rings
Throughout the text, let R be a commutative ring with identity (often called a commutative unital ring) and with multiplicative group of units R∗. Likewise, let f(x) = a0x +a1x n−1+ · · ·+an−1x+an be a polynomial of the variable x over R such that a0 ∈ R∗. Traditionally, R[x] is the ring of all polynomials of x over R; thereby f(x) ∈ R[x]. For an arbitrary but fixed element α, suppose f(x) is t...
متن کاملExtensions of Commutative Rings in Subsystems of Second Order Arithmetic
We prove that the existence of the integral closure of a countable commutative ring R in a countable commutative ring S is equivalent to Arithmetical Comprehension (over RCA0). We also show that i) the Lying Over ii) the Going Up theorem for integral extensions of countable commutative rings and iii) the Going Down theorem for integral extensions of countable domains R ⊂ S, with R normal, are p...
متن کاملExtensions of strongly alpha-reversible rings
We introduce the notion ofstrongly $alpha$-reversible rings which is a strong version of$alpha$-reversible rings, and investigate its properties. We firstgive an example to show that strongly reversible rings need not bestrongly $alpha$-reversible. We next argue about the strong$alpha$-reversibility of some kinds of extensions. A number ofproperties of this version are established. It is shown ...
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ژورنال
عنوان ژورنال: Topology and its Applications
سال: 2011
ISSN: 0166-8641
DOI: 10.1016/j.topol.2011.06.015