Last time For the rest of today, let R be a ring with 1. For A R,let A denote the smallest ideal containing A,called Ä p q the ideal generated by A,andisequalto I. I an ideal A I £Ñ An ideal I is principle if there is some a R for which I a . P “p q For example, in Z x , r s 2x, 3x x , but 2,x is not principle. p q“p q p q Note: 2,x is principle in Q x , because it contains a unit! p q r s Fact: If u is a unit, then ur r . p q“p q Some facts from last time: 1. An ideal I of R is equal to R if and only if I contains a unit. 2. If R is commutative, then AR RA RAR A .Further, “ “ “p q R is a field if and only if its only ideals are 0 and R. 3. If R is a field then any nonzero ring homomorphism from R into another ring is an injection. An ideal M in the ring R is a maximal ideal if M R and the ‰ only ideals containing M are M and R. More facts from last time: 1. In a ring with identity every proper ideal is contained in a maximal ideal. 2. Let R be a commutative ring. The ideal M is maximal if and only if R M is a field. { Generalizing the integers: Prime ideals Ring theory in Number theory. For example, read about the reduction homomorphism (p. 245), and its role in finding integer solutions to equations like x2 y2 3z2 or xn yn zn ` “ ` “ Definition Let R be a commutative ring. An ideal P is a prime ideal if P R ‰ and whenever ab P ,eithera P or b P . P P P Example:WhataretheprimeidealsinZ? Proposition Let R be a commutative ring. The ideal P is a prime ideal if and only if R P is an integral domain. { Corollary Let R be a commutative ring. Every maximal ideal of R is a prime ideal. The many kinds of rings Assume all rings R have 1 for a moment. We already know. Commutative rings: multiplication is commutative. Division rings: R 0 , is a group. p ´t u ˆq Fields: R 0 , is an abelian group. p ´t u ˆq Integral domains (or domains, or IDs): commutative and no zero divisors (cancellation works). New: the many kinds of (integral) domains. Principal ideal domains (PIDs): every ideal is principal. Unique factorization domains (UFDs): elements factor uniquely into primes. Euclidean domains (EDs): there’s a division (i.e. Euclidean) algorithm (you need a norm to determine remainders). You can show that Rings Comm. rings IDs UFDs PIDs EDs Fields t u à t u à t u à t u à t u à t u à t u Let R be a commutative integral domain with 1. Definition Let a, b R with b 0. P ‰ 1. a is a multiple of b if there exists x R with a bx.Wesay P “ b divides a, denoted b a. | 2. A greatest common divisor of a and b is a nonzero element d dividing a and b such that d a and d b implies d d. 1 | 1 | 1 | Well-defined?? “a” versus “the” Proposition Let a, b, d, d R. 1 P 1. Abusing parentheses: If the ideal generated by a, b is the t u same as the ideal generated by d,thend gcd a, b ,i.e. “ p q a, b d a, b d. p q“p qùñpq“ 2. Uniqueness: If d and d1 generate the same ideal, then d ud for some unit u R . 1 “ P ˆ In particular, if d and d1 are both greatest common divisors of a and b,thend ud. 1 “ Finding and using greatest common divisors The previous proposition said if a, b d then d gcd a, b . p q“p q “ p q Careful! Not every ideal is principal! So the converse is not always true! Ex: x, y in Q x, y . p q r s (Corollary) However, if R is a principal ideal domain (an integral domain where every ideal is principal), we have 1. the ideal a, b is the same as the ideal gcd a, b ,forany p q p p qq gcd a, b ,andso p q 2. since a, b ar bs r, s R ,thismeansthat p q“t ` | P u ar bs gcd a, b for some r, s R. ` “ p q P But what are r and s? To calculate, we need a division algorithm! Euclidean domains is where that happens, but that’s ok: every PID is a ED. Some more facts about PID’s Recall that a prime ideal P is an ideal satisfying if ab P ,thena P or b P . P P P Also, recall that maximal ideals are all prime ideals. Proposition In a PID, every non-zero prime ideal is a maximal ideal. Corollary If R is commutative and R x is a PID, then R is a field. r s.