TY - JOUR

T1 - Rings in which every prime ideal is contained in a unique maximal right ideal

AU - Sun, Shu-Hao

PY - 1992/4/6

Y1 - 1992/4/6

N2 - Our rings have identities and a pm ring is one having the property of the title. In an earlier paper (J. Pure Appl. Algebra 76 (2) (1991) 179-192), we discussed certain conditions on two-sided ideals, equivalent in the commutative case to the pm property. In the present paper we analyze similar conditions on one-sided ideals. As a consequence, we shall prove that, if the Prime Ideal Theorem (PIT) holds, then the right-ideal lattice IdrR of a ring R is normal if and only if R is Gelfand and satisfies MIT (the Maximal Ideal Theorem). This allows us to redefine Gelfand rings and to establish (in a later paper) Gelfand-Mulvey duality only using PIT rather than MIT.

AB - Our rings have identities and a pm ring is one having the property of the title. In an earlier paper (J. Pure Appl. Algebra 76 (2) (1991) 179-192), we discussed certain conditions on two-sided ideals, equivalent in the commutative case to the pm property. In the present paper we analyze similar conditions on one-sided ideals. As a consequence, we shall prove that, if the Prime Ideal Theorem (PIT) holds, then the right-ideal lattice IdrR of a ring R is normal if and only if R is Gelfand and satisfies MIT (the Maximal Ideal Theorem). This allows us to redefine Gelfand rings and to establish (in a later paper) Gelfand-Mulvey duality only using PIT rather than MIT.

UR - http://www.scopus.com/inward/record.url?scp=38249013172&partnerID=8YFLogxK

U2 - 10.1016/0022-4049(92)90096-X

DO - 10.1016/0022-4049(92)90096-X

M3 - Article

AN - SCOPUS:38249013172

SN - 0022-4049

VL - 78

SP - 183

EP - 194

JO - Journal of Pure and Applied Algebra

JF - Journal of Pure and Applied Algebra

IS - 2

ER -