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EXTREMAL ORDERS INSIDE SIMPLE ARTINIAN RINGS

In this paper, all rings are associative with a multiplicative unit and all modules are unitary. If A is a ring, J(A) will denote its Jacobson radical, U(A) its group of units, Z(A) its center, A* its set of nonzero divisors, and Mn(A) the ring of n n × matrices over A. The residue ring A/J(A) will be denoted by A . And Q denotes a simple artinian ring with finite dimension over its center Z(Q), while D denotes a division ring. In the second section we briefly discuss some of the ring theoretic properties and definitions. In the third section we will see that semihereditary V-orders are extremal V-orders and obtain a diagram of maximal V-orders when V is a Henselian valuation ring. In the fourth section we show that inside Bezout orders, extremal V-orders are precisely semihereditary, which is a generalization of Proposition 2.1 of [1]. In the last section we will examine the effect of Henselization on maximal and semihereditary orders.