Krull dimension and invertible ideals in Noetherian rings
T. H. Lenagan
1976
Proceedings of the Edinburgh Mathematical Society
In this note we consider the question: If R is a right Noetherian ring and / is an invertible ideal of R, how do the Krull dimensions of various modules, factor rings and over-rings of R, connected with /, compare with the Krull dimension of R1 This question is prompted by results in (5) and (6) . In comparing the Krull dimension of the ring R with that of the ring R/I, the best result would be that the Krull dimension of the ring R is exactly one greater than that of the ring R/I. This result
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... s not true in general; however, we see, in Theorem 2.4, that if the invertible ideal is contained in the Jacobson radical the result holds. In the general case we find it is necessary to introduce an over-ring T of R generated by the inverse 7 " 1 of/. We then see that the Krull dimension of R is the larger of two possibilities: (a) Krull dimension of R/T plus one or (b) Krull dimension of T. In order to prove this result we construct a strictly increasing map from the poset of right ideals of R to the cartesian product of the poset of right ideals of T with a poset of certain infinite sequences of right ideals of R/I. (2) and (4). We shall assume that each ring has a unit and that modules are unital. For background knowledge of Krull dimension we refer the reader to Definitions and preliminary results We recall first the definition of the deviation of a poset (partially ordered set). In (4) the definition was given only for positive integers, but, following Krause (3), we allow any ordinal number. Let E be a poset; if a, b e E, then [a, b~] = {x e E \ a g x g b] is again a poset. If E is a discrete set then dev E = -1. If E is Artinian (that is, each decreasing sequence of distinct elements of E is finite) then dev E = 0. Suppose that, for a given ordinal a, all posets Fwith dev F«x are known; then we define dev E = a if each decreasing sequence a u a 2 , ... of elements of E such that dev [a t+1 ,a{\4:u, for each/, is finite. If there is no ordinal a such that dev E = a we say that E has no deviation. This definition is applied to £P{M), the lattice of i?-submodules of an Rmodule M. The Krull dimension of M, denoted by \M\, is defined by | M | = dev (JS?(M)). In particular, | R \ = dev (£f(R R ))-Proposition 1. 1 (see (3) ). If M is a Noetherian module, then M has a Krull dimension. E.M.S.-20/1-F
doi:10.1017/s0013091500010580
fatcat:a5mwhtohk5ai7iqlkvs6fx7wxa