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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 ,
JAE KEOL PARK  and
S. TARIQ RIZVI
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 author.
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Abstract

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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 RR ⊆ ER, 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.

Keywords

Primary 16D5016S90Secondary 16L60
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 52 , Issue A: Rings and Modules in Honour of Patrick F. Smith's 65th Birthday , July 2010 , pp. 121 - 138
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.

References

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