Warm-up Let R R x , I xR x ,andJ x2R x .Recallthat “ r s “ r s “ r s ' : R x R defined by f x f 0 r sÑ p q fiÑ p q is a (ring) homomorphism. 1. Use the first isomorphism theorem to show that R I R (as { – rings). 2. Show that I J R. { fl 3. Explain why R J I J R. p { q{p { q– 4. Show that R J S,where { – xy S x, y R . “ 0 x P "ˆ ˙ˇ * ˇ [Hint: compare the results of p x q ˇx and p x q x with PQ ˇ and P Q,where p q p q p q` p q ` n p x a0 a1x anx , a0 a1 b0 b1 p q“ ` `¨¨¨` m and P ,Q . q x b0 b1x bmx , “ 0 a0 “ 0 b0 s p q“ ` `¨¨¨` ˆ ˙ ˆ ˙ Last time: isomorphism theorems Theorem 1. (First iso thm for rings)If' : R S is a homomorphism of Ñ rings, then ker' is an ideal of R, the image of ' is a subring of S and R ker' is isomorphic as a ring to ' R . { p q 2. If I is an ideal of R,thenthenatural projection R R I defined by r r I Ñ { fiÑ ` is a surjective ring homomorphism with kernel I.Thusevery ideal is the kernel of some ring homomorphism and vice versa. Theorem. Let R be a ring. 1. (Second isomorphism theorem for rings)LetA be a subring and B be an ideal of R.ThenA B a b : a A, b B ` “t ` P P u is a subring of R, A B is an ideal of A and X A B B A A B . p ` q{ – {p X q 2. (Third isomorphism theorem for rings)LetI and J be ideals of R with I J.ThenJ I is an ideal of R I and Ä { { R I J I R J. p { q{p { q– { 3. (Fourth isomorphism theorem for ring)LetI be an ideal of R. The correspondence A A I is an inclusion preserving Ø { bijection between the subrings A of R that contain I and the set of subrings of R I.Furthermore,A is an ideal of R if and { only if A I is an ideal of R I. { { Proof. These all hold for the underlying abelian (additive) groups. Remains to be checked: multiplication. Multiplication follows because the related isomorphisms naturally generalize as ring homomorphisms. Last time: Recall, an ideal of a ring R is a subring I R which satisfies Ñ rI I and Ir I for all r R. Ñ Ñ P To check that I is an ideal, check that 1. I is not empty (usually is 0 I?) P 2. I is closed under subtraction:foralla, b I,isa b I? P ´ P 3. I is closed under left and right multiplication by R: for all r R,isrI I and Ir R? P Ñ Ñ If I and J are ideals of R,thensoareI J, X I J i j : i I,j J ` “t ` P P u IJ finite sums of elements of the form ij i I,j J “t | P P u n I finite sums of elements of the form a1 ...an ai I “t | P u But in general, none of I J, ij i I,j J , a1 an a I Y t | P P u t ¨¨¨ | ` P u are ideals. Generating ideals Definition Let R be a ring and A R. Ä 1. Let A denote the smallest ideal containing A,calledthe p q ideal generated by A,andisequalto I. I an ideal A£I Examples: Ñ R 1 R p q“p q“ In Z x , x xZ x . r s p q“ r s (An ideal generated by one element is called a principal ideal) For ideals I,J,wehave I,J I J. p q“ ` 2. Let RA r1a1 rnan ri R, ai A, n Z` , “t `¨¨¨` | P P P u AR a1r1 anrn ri R, ai A, n Z` ,and “t `¨¨¨` | P P P u RAR r1a1r1 rnanr1 ri R, ai A, n Z` . “t 1 `¨¨¨` n | P P P u Proposition. If R is commutative, then AR RA RAR A . “ “ “p q Proposition Let R be a ring with 1, and let I be an ideal of R. 1. I R if and only if I contains a unit. “ 2. Assume R is commutative. Then R is a field if and only if its only ideals are 0 and R. Corollary If R is a field then any nonzero ring homomorphism from R into another ring is an injection. Maximal ideals Definition An ideal M in the ring S is a maximal ideal if M S and the only ‰ ideals containing M are M and S. Proposition In a ring with identity every proper ideal is contained in a maximal ideal. Theorem (Zorn’s Lemma) If is a non-empty partially ordered set in which every chain has A an upper bound, then has a maximal elemet. A Proposition Let R be a commutative ring. The ideal M is maximal if and only if R M is a field. { You try:LetI 3,x2 in Z x . “p q r s (a) Describe the set 3,x2 . Conclude that I is a proper ideal. p q (b) Show that I is not principle. [Hint: Suppose there were some a x Z x such that a x .Then p qP r s p p qq what does it mean that x2 a x ?that3 a x ?Arriveata Pp p qq Pp p qq contradiction.] (c) Show that I is not maximal. [Hint: Give an example of another proper ideal that contains I.] 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..