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Chapter XI Orders in Semisimple Artinian Rings

Elsevier B.V.
DOI: 10.1016/s0079-8169(08)60329-3
  • Mathematics


Publisher Summary The ring of rational integers is noetherian with the field of rationals as quotient field. From a ring theoretic standpoint, commutative fields are precisely the commutative simple artinian rings. It, therefore, is natural to generalize the idea of a quotient ring of an integral domain to noncommutative rings and then to attempt to characterize those rings with semisimple artinian quotient rings. This chapter presents a theorem that gives a characterization within R of those left ideals of R that occur as images of left ideals of Q under Φ. The chapter shows that Φ is an n-semilattice isomorphism. In the particular case that Q is a semisimple artinian ring, a complemented lattice of left ideals is obtained that is only a ∩-subsemilattice of the lattice of left ideals of R.

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