# Applied Group Theory for Chemists, Physicists and Engineers by Allen Nussbaum By Allen Nussbaum

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Extra info for Applied Group Theory for Chemists, Physicists and Engineers

Example text

It is elementary to show that any right Goldie domain has dimension 1. 14 THEOREM The following conditions on a ring R are equivalent: (1) R is semiprime right Goldie; (2) R is semiprime, Z(R) = 0 and dim R < oo; (3) R has a semisimple right quotient ring S. 14 until Chapter 4. There, the orders in simple Artin rings will be discussed at great length. The following concept plays a fundamental role both in the 28 Simple Noetherian rings study of orders in semisimple rings and the representation theory for simple rings.

B. 25(3) can be generated by 2 elements. Thus emerges an intimate connection between these rings and the classical Dedekind domains. 28 THEOREM (Webber ) Let A be a simple, right hereditary, right Noetherian ring. Let I, J and K be any three nonzero right ideals of A, J essential in A. Then, there exists a right ideal L suCh that I E8 K :::::; J E8 L. Proof If I :::::; J, we are finished. Otherwise, choose j 0 E J which is regular and set J 0 = j 0 I. Since I :::::; J 0 , if we set L 0 = K, I E8 K :::::; J 0 E8 L 0 • Simple Noetherian rings 36 Moreover, J 0 c J.

Next, for all r E R, tr - rt E I and has degree less than n. ). Moreover, if t = an + rn-lan-l + ... + ro, tr = rt Vr E R (I) implies that by equating the coefficients of both sides of (I). Thus, if n =F 0, ais inner, contradicting (b). ). , a(r) = ar - ra, Vr E R. ) . ). 1 1E I} Thus, (a) ~ (b). 2a THEOREM Suppose R is a commutative domain with the characteristic of R, char R, = p 0. )R is simple iff R is a field and dimcR is infinite, where C = constants a = {r E R I a(r) = 0}. Proof ~: First, a(rP) = 0 for all (nonunits) r E R.