Extensions of Commutative Rings in Subsystems of Second Order Arithmetic
نویسنده
چکیده
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 provable in WKL0.
منابع مشابه
On the commuting graph of non-commutative rings of order $p^nq$
Let $R$ be a non-commutative ring with unity. The commuting graph of $R$ denoted by $Gamma(R)$, is a graph with vertex set $RZ(R)$ and two vertices $a$ and $b$ are adjacent iff $ab=ba$. In this paper, we consider the commuting graph of non-commutative rings of order pq and $p^2q$ with Z(R) = 0 and non-commutative rings with unity of order $p^3q$. It is proved that $C_R(a)$ is a commutative ring...
متن کاملSecond order theories with ordinals and elementary comprehension
We study elementary second order extensions of the theory ID 1 of non-iterated inductive deenitions and the theory PA of Peano arithmetic with ordinals. We determine the exact proof-theoretic strength of those extensions and their natural subsystems, and we relate them to subsystems of analysis with arithmetic comprehension plus 1 1 comprehension and bar induction without set parameters.
متن کاملArithmetic Deformation Theory of Lie Algebras
This paper is devoted to deformation theory of graded Lie algebras over Z or Zl with finite dimensional graded pieces. Such deformation problems naturally appear in number theory. In the first part of the paper, we use Schlessinger criteria for functors on Artinian local rings in order to obtain universal deformation rings for deformations of graded Lie algebras and their graded representations...
متن کاملThe Auslander-Reiten Conjecture for Group Rings
This paper studies the vanishing of $Ext$ modules over group rings. Let $R$ be a commutative noetherian ring and $ga$ a group. We provide a criterion under which the vanishing of self extensions of a finitely generated $Rga$-module $M$ forces it to be projective. Using this result, it is shown that $Rga$ satisfies the Auslander-Reiten conjecture, whenever $R$ has finite global dimension and $ga...
متن کاملOn the commuting graph of some non-commutative rings with unity
Let $R$ be a non-commutative ring with unity. The commuting graph of $R$ denoted by $Gamma(R)$, is a graph with a vertex set $Rsetminus Z(R)$ and two vertices $a$ and $b$ are adjacent if and only if $ab=ba$. In this paper, we investigate non-commutative rings with unity of order $p^n$ where $p$ is prime and $n in lbrace 4,5 rbrace$. It is shown that, $Gamma(R)$ is the disjoint ...
متن کامل