Torsion theories and rings of quotients of Morita equivalent rings

A Ring of Quotients for Group Rings which is Easy to Describe

“…Luedeman remarks that the method of Utumi for ordinary injectivity does not readily generalize; however that of Kaye [3] does. More generally Turnidge [6] has studied the connections between idempotent topologizing families in Morita equivalent rings. If G\ R JK-+ S J% and H\ s J(-> R Jt are functors giving a category equivalence, there is a pairing between the hereditary torsion theories [6] in R <J( and $J£ and, hence, between the idempotent topologizing families of left ideals.…”

“…More generally Turnidge [6] has studied the connections between idempotent topologizing families in Morita equivalent rings. If G\ R JK-+ S J% and H\ s J(-> R Jt are functors giving a category equivalence, there is a pairing between the hereditary torsion theories [6] in R <J( and $J£ and, hence, between the idempotent topologizing families of left ideals. Let ^(R) be a torsion theory in R J( then the pairing is given by corresponding to ^(R) the torsion class $~{S)= {Me 8 J£ | H(M) e3T(R)}.…”

Torsion Theories and Semihereditary Rings

“…(See Gabriel [6] and the Walkers [13].) The ring of quotients of the integers Z with respect to the strongly complete Serre class of torsion abelian groups is the field of rationals Q and the kernel of the mapping rjG: G->Q ®zG defined by i)a(g) = l®g is the torsion subgroup T(G) of G. In general if (3, fJ) is a torsion theory for sSJi, the kernel of the mapping [January rjM: M-*Q3®rM defined by r]M(m) = 1 ®m is contained in the torsion submodule T(M) of M. (See [12,Proposition l.l].) It can be shown that whenever ker rjM = T(M) for every left P-module M the ring Q3 is flat as a right P-module.…”

“…(See [12,Theorem 1.4].) For example given any left Ore ring R, the ring of left quotients of R with respect to the torsion theory studied by Levy is the classical ring of left quotients of R which is flat as a right P-module.…”