Hereditary ring

Abstract: Let A = A(k) be the first Weyl algebra over an infinite field k, let P be any noncyclic, projective right ideal of A and set S = End(P). We prove that, as k-algebras, S f A. In contrast, there exists a noncyclic, projective right ideal Q of S such that S End(Q). Thus, despite the fact that they are Morita equivalent, S and A have surprisingly different properties. For example, under the canonical maps, Autk(A) Pick(A) Pick(S). In contrast, Autk(S) has infinite index in Pick(S). Introduction. Given a comnmutative domain R, the (first) Weyl algebra A(R) is defined to be the associative R-algebra (thus R is central subring) generated by elements x and y subject to the relation xy yx = 1. When no ambiguity is possible, we write A for A(R). Let F be a field of characteristic zero. Then A = A(F) is a simple ring and, indeed, may be thought of as one of the nicest and most important examples of simple Noetherian rings. The initial motivation for this paper was the following result of Smith. If P is the noncyclic right ideal P = x2A + (xy + 1)A of A then End(P) t A (as F-algebras) [11]. Note that, as char(F) = 0, A is a simple hereditary ring and so End(P) is automatically Morita equivalent to A. Even worse, as PEDP -ADA [17], the full matrix rings M2(A) and M2(End(P)) are isomorphic. Thus any proof of Smith's result must be fairly subtle and may therefore provide useful invariants for A. The first main aim of this paper is to generalize Smith's result to an arbitrary projective right ideal of A. While the proof of this is harder than that of Smith's result it does provide a more informative proof in the sense that, unlike Smith, we do not (and cannot) require an explicit description of End(P). For the rest of this introduction k will denote a field of arbitrary characteristic and all isomorphisms of rings will be k-algebra isomorphisms. THEOREM A. Let P be a projective right ideal of A = A(k). Then End(P) A if and only if P is a cyclic right ideal of A. (See Theorem 3.1.) This has some easy corollaries: COROLLARY B. Let P and Q be projective right ideals of A = A(k). Then End(P) End(Q) if and only if P = to(Q) for some t E D(k), the division ring of fractions of A, and a E Autk (A), the group of k-automorphisms of A. Received by the editors February 13, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 16A65, 16A72, 16A19. Supported in part by an NSF grant. (?)1987 American Mathematical Society 0002-9947/87 $1.00 + $.25 per page

Abstract: A ring (with unit element) Λ is called semi-primary if it contains a nilpotent two-sided ideal N such that the residue ring Γ = Λ/N is semi-simple (i.e. l.gl.dim Γ = r.gl.dim Γ = 0). N is then the (Jacobson) radical of Λ. Auslander [1] has shown that if Λ is semi-primary then The common value is denoted by gl. dim Λ. On the other hand, for any ring Λ the following conditions are equivalent : (a) 1. gl. dim Λ ≦ 1, (b) each left ideal in Λ is projective, (c) every submodule of a projective left Λ-module is projective. Rings satisfying conditions (a)-(c) are called hereditary. For integral domains the notions of “hereditary ring” and “Dedekind ring” coincide.

Abstract: Let A be a semi-primary ring i.e. its (Jacobson) radical N is nilpotent and Γ = A/N is an Artinian ring. The problem of characterizing a semi-primary ring A all of whose residue rings have finite global dimension—was dealt in several papers. It turns out that A is such a ring if and only if A is a residue ring of a semi-primary hereditary ring Ω. It was suggested that Ω is uniquely determined up to an isomorphism by the condition Ω/M & A/N, where M is the radical of Ω. One can prove that if A is an epimorphic image of a semi-primary hereditary ring Ω, then Ω is uniquely determined (up to an isomorphism) by the conditions (a) Ω admits a (semi direct sum) splitting, Ω = Γ + A + M and (b) Ω/M TM A/N. The following ring furnish a counter example to the uniqueness statement if we don't assume condition (a), even if A admits a splitting. Let fc be a field of characteristic p Φ 0, and let a; be a transcendental element over k. Let R = k(x) ® A { s ) k(x ) and let V be the radical of R. Then V contains the nonzero element x ® 1 — 1 ® x. Let 2 be a subring of the 3 x 3 matrix algebra over R, which consists of all matrices M for which:

Abstract: A ring R is said to be right (left) hereditary if every right (left) ideal in R is projective, that is, a direct summand of a free R -module. Cartan and Eilenberg [3, p. 15] ask whether there exists a right hereditary ring which is not left hereditary. The answer: yes.