By Lawrence S. Levy
The direct sum behaviour of its projective modules is a basic estate of any ring. Hereditary Noetherian top earrings are probably the single noncommutative Noetherian earrings for which this direct sum behaviour (for either finitely and infinitely generated projective modules) is well-understood, but hugely nontrivial. This publication surveys fabric formerly on hand purely within the examine literature. It presents a re-worked and simplified account, with superior readability, clean insights and lots of unique effects approximately finite size modules, injective modules and projective modules. It culminates within the authors' strangely entire constitution theorem for projective modules which consists of autonomous additive invariants: genus and Steinitz type. numerous purposes reveal its software. the idea, extending the well known module concept of commutative Dedekind domain names and of hereditary orders, develops through an in depth examine of straightforward modules. this depends upon the sizeable account of idealizer subrings which varieties the 1st a part of the publication and gives an invaluable basic building device for attention-grabbing examples. The publication assumes a few wisdom of noncommutative Noetherian earrings, together with Goldie's theorem. past that, it really is mostly self-contained, because of the appendix which gives succinct money owed of Artinian serial earrings and, for arbitrary jewelry, effects approximately lifting direct sum decompositions from finite size pictures of projective modules. The appendix additionally describes a few open difficulties. The heritage of the themes is surveyed at acceptable issues
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Additional info for Hereditary Noetherian prime rings and idealizers
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Thus S/B S ∼ and hence B S is isomaximal of type U . Now, T B = T =U since B is generative in T ; hence SB S = ST B S = S and so B S is generative in S. Thus A = B S ∈ α1 . Finally, each simple submodule X of T /B is isomorphic to U , and hence not to V . 14, B S ∩ T = B ; that is, (A )↓T = B . The ‘inclusion-preserving’ statement is obvious in this case. Case i = 2. Here, (A )↓T = A ∩ T and (B )↑S = B S. Choose A ∈ α2 and let B = A ∩ T . Since A ⊃ A we have B ⊇ A ∩ T = A. Moreover, the inclusion is proper since A = B implies A = B S = (A ∩ T )S = A , contradicting A ∈ α2 .
Ut . Then ti=1 λ(Ui )R = n + 1. (iv) Each simple R-module arises as an R-composition factor of some unique simple S-module. 5. Notation. In the light of (iv) above, given any iterated basic idealizer R from a ring S, and given a simple R-module V , we let V ∧ denote the unique (isomorphism class of) simple S-module of which it is an R-composition factor. It is, of course, possible that V is a simple S-module, in which case V ∧ = V . Proof. We proceed by induction on n, with the case n = 1 being covered by §3.
Note that T ⊆ T . 7, each of UT and UT is uniserial of length 2; say with composition factors V, W and V , W respectively from top to bottom. Further UT embeds fully in UT . We now wish to apply the induction hypothesis to the two rings T and T and the simple module WT . 4, T /A ∼ = W which is simple. = W (t) ). Moreover W ⊗T T ∼ So the hypotheses on the rings and simple module are as required. 5, B = {B2 ⊂ . . ⊂ Bn } is a basic chain in T and has the same rank sequence as {A2 ⊂ . . ⊂ An }; and IT (B) = IS (A) = R.