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

OSOFSKY, BARBARA L. ; PARK, JAE KEOL ; RIZVI, S. TARIQ

Glasgow mathematical journal, Tome 52 (2010) no. A, pp. 121-138

Résumé

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.

Détail

DOI : 10.1017/S0017089510000248

Mots-clés : Primary 16D50, 16S90, Secondary 16L60

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