On quasihereditary rings

Burgess, W. D.; Fuller, K. R.

doi:10.1090/S0002-9939-1989-0964453-8

On quasihereditary rings
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by W. D. Burgess and K. R. Fuller PDF

Proc. Amer. Math. Soc. 106 (1989), 321-328 Request permission

Abstract:

The notion of a quashereditary semiprimary ring was introduced by Cline, Parshall and Scott and was studied extensively by Dlab and Ringel. It is known that if $R$ is semiprimary of global dimension $\leq 2$ then it is quasihereditary and that there is a serial ring of global dimension 4 which is not. This paper establishes three principal results: If $R$ is quasihereditary then the Cartan determinant conjecture is true for $R$ (i.e., $C(R) = 1$); a serial ring is quasihereditary iff it has a heredity ideal; in particular, every serial ring of global dimension 3 is quasihereditary; and there is a ring (a $0$-relations algebra in fact) of global dimension 3 which is not quasihereditary.

References

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Heredity ideals

Every semiprimary ring is the endomorphism ring of a projective module over a quasihereditary ring

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