4400algebra1

# 4400algebra1 - ALGEBRA HANDOUT 1 RINGS FIELDS AND GROUPS...

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Unformatted text preview: ALGEBRA HANDOUT 1: RINGS, FIELDS AND GROUPS PETE L. CLARK 1. Rings Recall that a binary operation on a set S is just a function ∗ : S × S → S : in other words, given any two elements s 1 , s 2 of S , there is a well-defined element s 1 ∗ s 2 of S . A ring is a set R endowed with two binary operations + and · , called addition and multiplication, respectively, which are required to satisfy a rather long list of familiar-looking conditions – in all the conditions below, a , b , c denote arbitrary elements of R – (A1) a + b = b + a (commutativity of addition); (A2) ( a + b ) + c = a + ( b + c ) (associativity of addition); (A3) There exists an element, called 0, such that 0 + a = a . (additive identity) (A4) For x ∈ R , there is a y ∈ R such that x + y = 0 (existence of additive inverses). (M1) ( a · b ) · c = a · ( b · c ) (associativity of multiplication). (M2) There exists an element, called 1, such that 1 · a = a · 1 = a . (D) a · ( b + c ) = a · b + a · c ; ( a + b ) · c = a · c + b · c . Comments: (i) The additive inverse required to exist in (A4) is unique, and the additive inverse of a is typically denoted − a . (It is easy to check that − a = ( − 1) · a .) (ii) Note that we require the existence of a multiplicative identity (or a “unity”). Every once in a while one meets a structure which satisfies all the axioms except does not have a multiplicative identity, and one does not eject it from the club just because of this. But all of our rings will have a multiplicative identity. (iii) There are two further reasonable axioms on the multiplication operation that we have not required; our rings will sometimes satisfy them and sometimes not: (M ′ ) a · b = b · a (commutativity of multiplication). (M ′′ ) For all a ̸ = 0, there exists b ∈ R such that ab = 1. A ring which satisfies (M ′ ) is called – sensibly enough – a commutative ring . Example 1.0: The integers Z form a ring under addition and multiplication. Indeed they are “the universal ring” in a sense to be made precise later. Thanks to Kelly W. Edenfield and Laura Nunley (x3) for pointing out typos in these notes. 1 2 PETE L. CLARK Example 1.1: There is a unique ring in which 1 = 0. Indeed, if r is any element of such a ring, then r = 1 · r = 0 · r = (0 + 0) · r = 0 · r + 0 · r = 1 · r + 1 · r = r + r ; subtracting r from both sides, we get r = 0. In other words, the only element of the ring is 0 and the addition laws are just 0+0 = 0 = 0 · 0; this satisfies all the axioms for a commutative ring. We call this the zero ring . Truth be told, it is a bit of an- noyance: often in statements of theorems one encounters “except for the zero ring.” Example 1.n: For any positive integer, let Z n denote the set { , 1 ,...,n − 1 } ....
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4400algebra1 - ALGEBRA HANDOUT 1 RINGS FIELDS AND GROUPS...

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