**Unformatted text preview: **05/12/2003 MON 16:26 FAX 6434330 MOFFITT LIBRARY .001 George M. Bergman Fall 2002, Math 113, Sec. 5 13 Dec., 2002
9 Evans Hall Final Examination 12:30—13:30 1. (12 points, 4 points each.) Complete the following deﬁnitions. (a) If X is a set, then a permutation of X means (b) If R and S are commutative rings, then a map f : R —) S is called a homomorphism
of commutative rings if (c) A proper ideal I of a commutative ring R is said to be a maximal ideal if 2. (36 points; 4 points each.) For each of the items listed below, either give an example,
or give a brief reason why no example exists. (If you give an example, you do not have to
prove that it has the property stated. Examples should be speciﬁc for full credit; i.e., even
if there are many objects of a given sort, you should name one.) (a) An element o-E S6 such that 0' (1 2 3) 0‘1 = (2 4 6). (b) An injective (i.e., one-to—one) homomorphism f : Z —> lRX. (Recall that le denotes
the group of nonzero real numbers under multiplication.) (c) A factorization of the polynomial 3 x3 + 29 x2 — 4x — 2 as a product of two
polynomials of lower degree in ©[x]. (d) A ring R, an ideal I g R, and elements a at b of R such that a+I = b+I.
(e) A polynomial f(x) E(D[x] which has no root in (D, but which is reducible in ©[x].
(f) A ﬁeld with exactly 100 elements. (g) An ideal I g Z[x] such that Z[x]/I is isomorphic to Z[i], the ring of Gaussian
integers. (h) Two elements of ZS [x] that are associates, but are not equal.
(i) A unique factorization domain which is not a principal ideal domain. 3. Short proofs. (22 points : 6+8+ 8.) (a) If f: X —> Y and g : Y —> Z are set maps such that the composite map
30f: X—> Z is injective, Show that f is injective. (b) Suppose a group G acts on a set X, and let S = {06 G | V x6 X, ox = x}. Show
that S is a normal subgroup of G. (You must show both that it is a subgroup and that it
is normal.) (0) Suppose P1 2 P2 2 2 P” 2 PH” 2 is a decreasing sequence of prime ideals.
Show that the ideal I = ﬁnal Pn is prime. (Here you are to take for granted that I is
an ideal, in contrast to the homework problem this is taken from, where you had to prove both that it was an ideal and that it was prime.) ...

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