I am in Reiss 258. See below for office hours on May 9 and May 10. For background and grading info see course mechanics .
The text is "Contemporary Abstract Algebra" by J. Gallian, 5th edition.
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May 23, 2006
Here's the final fun final . And here is the key for the final .
Everyone is expected to attend class regularly.
For previous problems and some solutions, go to the old homework page.
The final will be held on Thurs. May 11, from 12:30 to 2:30 pm in Reiss 264. I'll have office hours on Tues. May 9 from 2:30 to 4:30 and on May 10 from 4 to 6 pm. Note that as I told you in e-mail, there are copies of some useful preparatory material (on lattices) on top of the file cabinet just outside my office - Reiss 258.
Recall that a poset (P, >=) is a nonempty set P and a relation
>= on P which is P_1: anti-symmetric (a >= b and b >= a is equivalent
to a = b) and P_2: transitive. A poset is a latice provided that for
every two elements in the poset, there is a greatest lower bound
and a least upper bound - i.e., for all a,b in P, if there exists g in
P with g <= a, g <= b, and if h <= a, h <= b holds for any h in P,
then h <= g, and we call g the glb(a,b), denoted g = a /\ b (written
like intersection in the printed notes. Similarly, if there is a least upper
bound lub(a,b), it is denoted a \/ b and is like union for subsets.
These operations of meet and join as they are normally called
satisfy the four equations L_1, ... ,L_4 in the notes (join and meet are
required to be commutative, associative, idempotent, and to satisfy a
compatability condition corresponding to the idea that the join is bigger
than its components while the meet is smaller.
Conversely if a nonempty set L has two operations of these types
satisfying the four equations, then there is a unique partial order on L
which determines a lattice with meet and join the given operations.
Indeed, define
a <= b if and only if a \/ b = b.
Moreover, a \/ b = b if and only if a /\ b = a. If you take the
example of subsets of a given set, then join is union and meet is
intersection.
A poset (P, >=) has a zero-element 0 if
for all a in the poset, 0 less-or-equal a, and a
one-element 1 in the poset satisfies a less-or-equal 1
- in symbols,
0 <= a <= 1
Note that any two zero-elements are the same, and
similarly, any two one-elements are the same.
A poset (P, >=) with 0 and 1 is called complemented if
for every a in P, there exists a' in P such that
a \/ a' = 1, a /\ a' = 0
For instance, for subsets of S, 1 = S, 0 = empty-set, and
a' means the complement of a
a' = S - a.
Note that (i) the complement a' is uniquely determined, (ii) the
complement of a meet is the join of the complements:
(a /\ b)' = a' \/ b' (and vice versa)
and (iii) (a')' = a.
A lattice is called distributive provided that for all a,b,c
D: (a \/ b) /\ c = (a /\ c) \/ (b /\ c)
and the same form of equation holds with join and meet reversed.
The lattice of all subsets of a fixed set S does satisfy both the
distributive and complemented properties.
Exercise: Generalize the last exercise of the handout, which asked
you to prove that for subsets of S, letting ``I'' denote intersection,
``U'' mean union, and ``-'' denotes set-theoretic difference,
(A U B) - (A I B) = (A - B) U (B - A).
For complemented lattices, this equation reads:
(a \/ b) /\ (a /\ b)' = (a /\ b') \/ (b /\ a')
Can you now prove this equation for complemented distributive
lattices using property D and the properties of complement?
What is the connection of these lattices with logic?
As I said in class, the final will be optional but recommended. Your grade will not be lowered if you do badly but, if you do well, it will cover a multitude of sins ;-) Of course, if you have done little through the course, doing very well on the final isn't going to get you an optimum grade but it can make up for a lot. Further, the final will be "de novo" so it won't require anything more than an open mind, an enthusiastic attitude, and a general familiarity with the algebraic method. It will give you a chance to learn some really neat results!
Here are some problems to try for tomorrow and Monday:
New ones: Show that for any ring R, the set I of all elements which have finite additive order constitute an ideal. (x in R has finite additive order if there exists a pos. integer n such that nx = 0; let n_x be the least such pos. integer for a given x in R. Show that n_(x-y) is at most the l.c.m. of n_x and n_y, etc.)
Let phi be an epimorphism of rings phi: R --> S. SHow that if R has a unit element 1 for multiplication phi(1) is a unit element for S. Is this true without the assumption of onto? (Recall that an epimorphism is a homomorphism which is onto.)
Let R be a commutative ring with 1 with only {0} and R as ideals. SHow that R must be a field.
In all of the following, suppose that R is a commutative ring with 1. (1) If A is an ideal of R and I is an ideal of A, must I be an ideal of R? (2) If S is a subring of R and I is an ideal of R with I contained in S, must I be an ideal of S? (3) If S is a subring of R and I is an ideal of R, must the intersection of I and S be an ideal of S? (4) If I and A are ideals of R, must the intersection of I and A be an ideal of R? (5) If A is a principle ideal of R and I is a principle ideal of A, must I be a principle ideal of R? (6) If I and A are maximal ideals of R, must the intersection of I and A be a maximal ideal of R? (7) When is (0) a prime ideal of R? For each problem 1 through 6, at least decide whether you think the answer should be Yes or No and, hopefully, give an argument or example. These will be discussed in class; some of you will be asked to do one at the board.
The following problems will make a good review of chapters 12 through 17. Don't forget that many problems can be viewed as special cases of theorems in the text.
We will talk about these problems tomorrow (4/19) and I may ask some of you to put them on the board.
For Algebra, here is the assignment: Let R and S be
rings and let R plus-in-a-circle S be the direct sum
(we'll write it as R .+. S here)
i.e., the ring with cartesian product of R and S as
its underlying set and with both operations defined
coordinatewise as usual. (See p. 231 in the text.)
(1) If R and S are integral domains,
must R .+. S also be an I.D.?
Either prove or disprove.
(2) Let R be a commutative ring with 1.
Show that if A is a maximal ideal in R,
then A is also a prime ideal in R.
(3) If R and S are rings and A is an ideal in R .+. S,
show that
(R .+. S)/A is isomorphic to (R/I) .+. (S/J)
for some I and J ideals in R and S. Also,
identify what I and J are in this case.
(4) Show that (x) is not a maximal ideal in Z[x],
but _is_ a maximal ideal in Q[x].
(earlier I had written this with angle-brackets which unfortunately
are ignored by the browser so the particular ideal referred to here
was previously hidden - sorry! and thanks to Ameet for noticing and
enquiring about a misprint. See below for the use of this notation
regarding the ideal (a) = {ar: r in R the ambient ring})
(5) Let a(x) and b(x) belong to Q[x], where Q denotes the
rational field. Let f(x) = x^2 + 1 in Q[x].
If f(x) divides a(x)*b(x), show that f(x) divides
either a(x) or b(x).
The homework for today (given in class on Friday) was #12 and #18 of chap. 17 and read ch. 18.
For homework due Fri. 4/7: Ch. 16: #26, 36. Also please read Ch. 17 up to p. 305 (skip the section on "weird dice"). Don't forget also to try that problem I mentioned: the equation
J = (phi(a)), where J is phi-inverse of I, I = (a),
(read "(a)" above as if it were "a in angle bracket",).
Homework due Wed.: Chapter 16: #18, 20, 24.
Homework due Friday: Let X be the set of all ordered pairs (a,b) with a and b in an I.D. D and b not equal to 0. Let a relation on X be defined by (a,b) ~ (c,d) if and only ad = bc. Show that the relation ~ is transitive - i.e., if (a,b)~(c,d) and (c,d)~(e,f), then (a,b)~(e,f). Note how your argument uses the properties of D. (I sketched in class the rest of the argument that ~ is an equivalence relation, and you should make sure you would know how to define the operations in the resulting quotient field, which is the set of equivalence classes determined by S/~. Could you show these operations are well-defined?
Homework due Mon. April 3: Ch. 15, #24,26,42 . Also please read Ch. 16.
Try Ch. 14, p. 262, #40; also in Ch. 15, pp. 277--281, 2,4,8,18,22,24,26,30,36,38,40,42, 46,54,56,58. Some of these will be selected as homework to be written up for Monday - so try a few to see if you have questions on Friday. Look through the xeroxed notes I gave you today in class for some possible hints.
For hw due for collection Mon. March 27, consists of the following four problems - though one of them involves several different parts and levels of abstraction.
From Ch. 13, #22 and #52 (pp. 247--249). First do #22 a,b,c as stated in the text. Then try to find the ``natural'' level of generality for (a),(b), and (c) - you'll need extra hypotheses for (c). What I mean is the following: Suppose S is some nonempty set. Let A be the set Fun(S,R) of functions from S to R, where R is a ring. Then one can add and multiply such functions element-wise: If f,g are in A, then define the function f + g by setting, for all x in S, (f+g)(x) = f(x)+g(x), where the first "+" is addition in A and the second "+" is addition in R - i.e., addition in A is defined in terms of addition in R. Similarly, define the product fg of f and g by (fg)(x) = f(x)g(x) for all x in S. Find conditions on R, as general as possible, so that (a) and (b) hold. Now find conditions so that (c) holds, too. The resulting rings will include the ring of real numbers as a special case.
Also do from Ch. 14, pp. 260--261: #4 and #24.
Here are two problems which will be part of the homework due on Mon. March 27. The remaining problems will be selected from ch. 14, pp. 260--263: #4,8,12,14,16,18,20,24,34,40,44,48,50. Try the problems from ch. 14 and we'll discuss some of them in class, and a few will be selected to be written up for homework. But the next two will definitely be included in the hw.
Read Ch. 13 and try 4,6,8,10,13,14,16,22. We will discuss these in class.
Homework for Monday March 20; for collection:
Ex 0. Show that in a ring, if there is a multiplicative inverse, it is unique. Ex 1. Show that in M_2(R), associativity holds by checking that (AB)C = A(BC) for the (1,2) position - I did the (1,1) case in class. If you weren't there, check the notes from someone who was. Ex 2. In an integral domain, if a is non-zero, must a^j be non-zero for j pos. integer. Prove or give a counter- example.
Homework due Friday Mar. 17 (for collection), Ch. 12, pp.235--236: #6,12,14,18,34 . Also, read Ch. 13.
For Monday, March 13 read chap. 12 (Intro to Rings) and try the following exercises on pp. 235 -- 237: # 2-12 (including the odd-numbers, too), 18,19,20,22,48. We'll discuss a few of them and then a few others will be selected for homework to be collected later. We will also review the midterm.
In class, we've talked about various topics in Chapters 6,7, 9, and 10. In ch. 6, be sure you could prove that the inner automorphisms form a group and are a subgroup of the group of all automorphisms of some fixed group G. In ch. 7, you should know the basic properties of cosets, Lagrange's theorem, and its Cor.1 through 4. We are skipping Ch. 8 (at least for now). For ch. 9, know the definition of a normal subgroup and theorems 9.1, 9.2, and 9.4. For Ch. 10, you should know the basic properties of homomorphisms: e.g., could you prove part 5 of Thm. 10.2? How are normal subgroups related to homomorphisms?
Remember also the simple exercises I suggested: Show that xH and yH are either disjoint or equal for all x,y in G when H is a subgroup. Also, check that defining xH * yH = (xy)H gives a group operation on the set G/H of cosets of H in G whenever H is a _normal_ subgroup. How does normality appear?
For homework for collection on Feb. 1 (Wed.) do the 2nd and 3rd parts of #16 of Chapter 2 ("socks and shoes"). We did the main part in class already; that is, (ab)^(-1) = b^(-1) a^(-1) but the other two parts ask for examples of elements and groups with certain properties. You should also be trying the problems in Chap. 3 now.
For Chap. 3, here are the problems I suggest working: 4,6,8,10,14--16,18--20,22,24,28,36,44,46,52. Note that these problems refer to the examples given on pp. 44--48 and pp. 59--63. You should also go back and read Chap. 1 now for the definition of dihedral group, which we will go over in class.
I'm listing problems for Chap. 4 and 5 now so that you can start trying them as soon as you are ready.
Chap. 4: #3--7,10,12,14,18,24,26,30,32,34,36,38,40,42,48, 50,54,56,60.62,64 (pp. 82--85).
Here are some hints. For #30, in any group, show that
if ab(a^-1) = b^-1, then (a^t)b(a^-t) = b for t even
= b^-1 for t odd
for t any nonnegative integer. This suffices for the
problem. Why? For #36, one direction is easy: If G is
cyclic, then G = for some a in G. Hence, if G is a
union of subgroups, one of the subgroups contains a and
hence G so that subgroup is not proper. For the other
half of the argument, assume that G is not cyclic and
then write G as a union of proper subgroups.
For #42, consider the case that G has no element of
infinite order; then do the other case.
Chap. 5: #2,4a,6,8,10,12,14,18,20,22,24,30,34,36,40,42,46, 52 (pp. 111--114).
Have fun!
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