Integrality in Associative Rings with

Since / has been chosen arbitrarily in L, we have shown that L C Bm(Ad(L)). Thus, the claim follows from Lemma 8.1.4. 

Note that the intersection of any two unitary subrings of D is a unitary subring of D. In particular, D possesses a uniquely determined smallest unitary subring. Let us (for the moment) denote this unitary subring by D. 

If D is finite, we call D the characteristic of D. Otherwise, we say that D has characteristic 0. 

Throughout this section, the letter D stands for an associative ring with 1. Let C be a unitary subring of D. 

An element d in D is called integral over C if there exist elements co,. .., cn i in C such that 

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In the second section, we shall look at commutative associative rings with 1. We prove that the set of all elements of a commutative associative ring D with 1 which are integral over a unitary subring of D forms a ring. 

A commutative associative ring D with 1 is called an integral domain if the product of any two elements in D 0 is in D 0. ... 

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