By Paul T. Bateman

I first taught an summary algebra direction in 1968. utilizing Hcrstein's themes in Algebra. it truly is not easy to enhance on his ebook; the topic can have develop into broader, with functions to computing and different parts, yet subject matters comprises the middle of any direction. regrettably, the topic hasn't develop into any more uncomplicated, so scholars assembly summary algebra nonetheless fight to profit the hot thoughts, particularly considering that they're most likely nonetheless studying tips to write their very own proofs.This "study consultant" is meant to assist scholars who're commencing to find out about summary algebra. rather than simply increasing the cloth that's already written down in our textbook, i made a decision to attempt to coach via instance, through writing out ideas to difficulties. i have attempted to decide on difficulties that will be instructive, and in numerous instances i have integrated reviews to aid the reader see what's particularly occurring. in fact, this learn advisor isn't really an alternative choice to an excellent instructor, or for the opportunity to interact with different scholars on a few demanding problems.Finally. i need to gratefully recognize the aid of Northern Illinois collage whereas scripting this research advisor. As a part of the popularity as a "Presidential educating Professor," i used to be given depart in Spring 2000 to paintings on tasks regarding instructing.

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That should give you the following equations. 435 = 1 · 377 + 58 377 = 6 · 58 + 29 58 = 2 · 29 gcd(435, 377) = gcd(377, 58) = gcd(58, 29) = 29 The repeated divisions show that gcd(435, 377) = 29, since the remainder in the last equation is 0. To write 29 as a linear combination of 435 and 377 we need to use the same equations, but we need to solve them for the remainders. 58 29 = = 435 − 1 · 377 377 − 6 · 58 Now take the equation involving the remainder 29, and substitute for 58, the remainder in the previous equation.

Show that Ann(a) is an ideal of R. 7. Let R be the ring Z2 [x]/ x4 + 1 , and let I be the set of all congruence classes in R of the form [f (x)(x2 + 1)]. (a) Show that I is an ideal of R. (b) Show that R/I ∼ = Z2 [x]/ x2 + 1 . (c) Is I a prime ideal of R? Hint: If you use the fundamental homomorphism theorem, you can do the first two parts together. 8. Find all maximal ideals, and all prime ideals, of Z36 = Z/36Z. 9. Give an example to show that the set of all zero divisors of a ring need not be an ideal of the ring.

Explain your answer. (b) Show that Z× 13 is a cyclic group. 2. Find all subgroups of Z× 11 , and give the lattice diagram which shows the inclusions between them. 3. Let G be the subgroup of 1 0 0 GL3 (R) consisting of all matrices of the form a b 1 0 such that a, b ∈ R . 0 1 Show that G is a subgroup of GL3 (R). 4. Show that the group G in the previous problem is isomorphic to the direct product R × R. × 5. List the cosets of the cyclic subgroup 9 in Z× 20 . Is Z20 / 9 cyclic? 6. Let G be the subgroup of GL2 (R) consisting of all matrices of the form m b 1 b , and let N be the subset of all matrices of the form .