By Jason P. Bell

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From xyd = 1 − d ∈ A it follows in the same way that yd ∈ A. ∗) Examples 5. e. has only ﬁnitely many maximal ideals, then it is obvious that R has large Jacobson radical. Thus every semilocal ring is convenient. Slightly more generally, if R is an integral extension of a semilocal ring R, then R has large Jacobson radical, hence is convenient. In particular the inﬁnite Galois extensions of R (cf. [K], there called inﬁnite “coverings” of R) are convenient rings, as well as the ﬁnite ones. 6. Assume there exists a family (ϕα : Rα → R | α ∈ I) of homomorphisms from convenient rings Rα to R such that R is the union of the subrings ϕα (Rα ).

This forces P (Aj ) = πj (P (A)). P (Aj ) = P (A). Since this holds for every j ∈ I, we conclude that i∈I We will say more about Pr¨ ufer hulls in II, §5 and in Part II of the book. ufer extensions and convenient §6 Examples of Pr¨ ring extensions In this section R is a ring and A is a subring of R. We are looking for handy criteria which guarantee that A is Manis or Pr¨ ufer in R, and we will discuss examples emanating from some of these criteria. 1. Assume that A is integrally closed in R. e.

If ϕ: A → B and ψ: B → C are ring homomorphisms and ϕ is ws then ψϕ(A) is ws in ψ(B). Proof. We have a commuting square with i an inclusion mapping and surjections p and q. Since ϕ and q are ws, the composite q ◦ ϕ = i ◦ p is ws. Thus also i is ws. 9. Let ϕ: A → B a ring homomorphism. ϕ is ws iﬀ ϕ(A) is ws in B. Proof. Applying Proposition 8 with ψ = idB we see that weak surjectivity of ϕ implies weak surjectivity of the inclusion mapping i: ϕ(A) → B. Conversely, if i is ws, then ϕ is ws, since ϕ = i ◦ p with p a surjection.

### Commutative algebra [Lecture notes] by Jason P. Bell

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