By Timofeenko A.V.

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**Example text**

Let a, b be non-zero, non-invertible elements in a onedimensional quasi-local domain R. Then some power of a is divisible by b. Proof. The ring R / ( b ) has exactly. one prime ideal, which is consequently nil (Theorem 25). The image of a in R / ( b ) is therefore nilpotent, i. , some power of a is divisible by b. + +- - + s = ym-1/(1 Theorem 109. Let the domain R b,e equal to V , n V2, where the V’s lie between R and its quotient Jield. Assume that each Vi is quasilocal, has maximal ideal Pi, and that Qi A R = Pi.

Proof. Let S denote the set of all elements ab in R where a p P and b p <(A). S is clearly a multiplicatively closed set. We claim that it is disjoint from I , for suppose that baA = 0, b p <(A), a p P. Then a A = 0, a E I C P, a contradiction. Enlarge I to a prime ideal Q , maximal with respect to disjointness from S. Then Q C <(A) and I C Q C P. By the minimality of P, we have Q = P and so P C <(A). By combining Theorems 80 and 81 we obtain a result that is among the most useful in the theory of commutative rings.

Any localization R s of a Noetherian ring is Noetherian. Proof. We know that any ideal in R S has the form IS with Z a suitable ideal in R. Since I is finitely generated, so‘is IS. Theorem 86. Let R be Noetherian and let A be a$nitely generated non-zero R-module with annihilator I,Let P be a prjme ideal in R minimal over I. Then P is the annihilator of a non-zero element of A. Theorem 83. Let R be a commutative ring, S a subring of R , and I an ideal of R contained in S. Suppose that I # S and that s - I c PI u - - - u P, where PI, .

### 2-Generator golod p-groups by Timofeenko A.V.

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