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PROPERTIES OF INJECTIVE HULLS OF A RING HAVING A COMPATIBLE RING STRUCTURE*

Published online by Cambridge University Press:  24 June 2010

BARBARA L. OSOFSKY
Affiliation:
Department of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ 08904-8019, USA e-mail: osofsky@math.rutgers.edu
JAE KEOL PARK
Affiliation:
Department of Mathematics, Busan National University, Busan 609-735, South Korea e-mail: jkpark@pusan.ac.kr
S. TARIQ RIZVI
Affiliation:
Department of Mathematics, Ohio State University, Lima, OH 45804-3576, USA e-mail: rizvi.1@osu.edu
Corresponding
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Abstract

If the injective hull E = E(RR) of a ring R is a rational extension of RR, then E has a unique structure as a ring whose multiplication is compatible with R-module multiplication. We give some known examples where such a compatible ring structure exists when E is a not a rational extension of RR, and other examples where such a compatible ring structure on E cannot exist. With insights gleaned from these examples, we study compatible ring structures on E, especially in the case when ER, and hence RRER, has finite length. We show that for RR and ER of finite length, if ER has a ring structure compatible with R-module multiplication, then E is a quasi-Frobenius ring under that ring structure and any two compatible ring structures on E have left regular representations conjugate in Λ = EndR(ER), so the ring structure is unique up to isomorphism. We also show that if ER is of finite length, then ER has a ring structure compatible with its R-module structure and this ring structure is unique as a set of left multiplications if and only if ER is a rational extension of RR.

Type
Research Article
Copyright
Copyright © Glasgow Mathematical Journal Trust 2010

Footnotes

*

To Patrick Smith on his 65th birthday, with thanks for all he has done for ring theory.

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PROPERTIES OF INJECTIVE HULLS OF A RING HAVING A COMPATIBLE RING STRUCTURE*
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