Transcription of RING THEORY 1. Ring Theory - Northwestern University
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RING THEORY 1. Ring Theory - Northwestern University
CHAPTER IVRING THEORY1. ring TheoryAringis a setAwith two binary operations satisfying the rules given below. Usually one binary operationis denoted + and called addition, and the other is denoted by juxtaposition and is called multiplication. The rules required of these operations are:1)Ais an abelian group under the operation + (identity denoted 0 and inverse ofxdenoted x);2)Ais a monoid under the operation of multiplication ( , multiplication is associative and there is atwo-sided identity usually denoted 1);3) the distributive laws(x+y)z=xy+xzx(y+z)=xy+xzhold for allx,y, one does not require that a ring have a multiplicative identity. The word ring may also beused for a system satisfying just conditions (1) and (3) ( , where the associative law for multiplicationmay fail and for which there is no multiplicative identity.
32 IV. RING THEORY If A is a ring, a subset B of A is called a subring if it is a subgroup under addition, closed under multiplication, and contains the identity. (If A or B does not have an identity, the third requirement would be dropped.) Examples: 1) Z does not have any proper subrings. 2) The set of all diagonal matrices is a subring ofM n(F). 3) The set of all n by n matrices which …
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