(a) Mathematical induction and applications to the properties of integer numbers, as the Euclidean Algorithm, Greatest Common Divisor, Unique Factorization: pp 2-9.

(b) Equivalence relations and classes: pp. 12-13.

(c) Definition of a group, examples (we only restricted ourselves to commutative groups), generators, cyclic groups: pp. 16-23.

(d) Homomorphisms, kernel, image (again, only for commutative rings), Theorems 3.2 and 3.3: pp. 33-36, 38.

(e) Cosets (no normal subgroups - in the commutative case we are concerned with all subgroups are automatically normal and we never introduced this concept, no right and left cosets for the exact same reason), index of a subgroup (Theorem 4.3), group of cosets, cacnonical homomorphism G --> G/H, Theorem 4.4 and Corollary 4.7: pp. 41-44, 46.

(f) We discussed the structure of cyclic groups and finite abelian groups (the latter without proof); all you need to remember is that (i) any cyclic group of order n is isomorphic to Z/nZ and (ii) any finite abelian group is isomorphic to the direct sum of cyclic groups of order a power of some prime number. In particular, any two cyclic groups of the same order are isomorphic and any finite abelian group is isomorphic to the direct sum of cyclic groups: pp. 55-56, 67-68.

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Lecture on 03.03.05: We introduced the concept of rings. We only consider commutative rings. Several examples have been mentioned (Z, R, C, the ring of continuous fiunctions on [0,1]). Intergal rings - not all rings are integral. This corresponds to pages 82-86 (not everything from the text was discussed in the class).

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Lecture on 03.05.05: We introduced fields. Examples were given. There is no intermediate field between Z and Q, except Q itself, but there are infinitely many intermediate subrings between Z and Q, constructed this way: fix any subset X of the set (finite or infinite) of all prime numbers and consider the rational numbers whose indecomosable representations only have powers of prime numbers from X in their denumenator. These rational numbers constitute a ring under the usual number-theoretical addition and multiplicaiton. Can you prove that these are the only subrings between Z and Q? The same pages in the text as for the previous lecture.

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Lecture on 03.08.05: We introduced ideals (only commutative case, no left and right ideals), the sum and product of two ideals. pp. 87-88.

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Lecture on 03.12.05: We intruduced ring homomorphisms, quotient rings by ideals (examples: Z/3Z is finite field and Z/4Z is finite ring which is not integral). There are only injective ring homomorphisms from any field to any ring. pp.90-92. Example 3 on p. 93.

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Lecture on 03.15.05: We proved that Z/nZ is a field if and only if n is a prime natural number. We also introduced the group of units U(R) for arbitray commutative ring and computed it in the special cases R=Z/10Z and Z/12Z. No new pages.

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Lecture on 3/17/04. We showed that there is a unique homomorphism from Z to any ring R, namely the homomorphism that maps an integer number n to n times the unit element of R. When the image of this homomorphism is an infinite set we get a ring injective homomorphism. If R is an integral ring and the image of Z in it is an infinite set we say that R has characteristic 0. If R is an integral ring and the image of Z in it is a finite set then this image is isomorphic to a finite field of type Z/pZ for some prime p. This prime number is called the characteristic of R. By the same token we have defined characteristic of arbitrary integral ring. pp 94,95.

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Lecture on 3/19/04. We showed how to construct a field K out of any integral ring R. The field K is called quotient field of R, or better the field of fractions of R. (Important special case is the field Q which is the fracion field of Z). An integral ring embeds in its fraction field, pp. 100-104.

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Lecture on 3/29/04. We introduced the ring of polynomials R[X] over arbitrary ring R. This was done via certain explicit construction using finite sequences of elements of R. pp. 195-108.

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Lecture on 3/31/04. We went over the Homework #9 problems. THe detailed solutions are already posted.

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Lecture on 4/2/04. Eucledean Algorithm for polynomials over a field, maximal number of roots of a polynomial of degree n. pp. 114-115.

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Lecture on 4/5/04. We showed that every iddeal in K[X] is principal (i. e. generated by a single elements). As a corollar, we showed that any two polynomails have a greatest common divisor. pp.118-120

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Lecture on 4/7/04. We went obver the problems in the homework. The detailed solutions are already posted.

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Lecture on 4/9/04. We proved Gauss Lemma (p. 137) and Gauss Theorem (p. 139), and talked about a generalization of the Gauss lemma to arbitrary rings. pp. 136-139.

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Lecture on 4/12/04. Click here.

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Lecture on 4/14/04. We completed the proof of the fact that every PID is UFD. We mentioned Gauss Theorem that if R is a UFD then so is the polynomial ring R[t]. We observed that the subring of F[t] (F arbitrary field) consisting of the polynomials that do not have a linear part is an integral ring which is not a UFD. pp. 143-148 (for the last two lectures).

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Lecture on 4/16/04. We introduced polynomial rings in many variables R[t_1,...,t_n], obtained by iteration of taking polynomial rings in one variable. We then considered degrees of polynomials in many variables and observed that if R is a UFD then so is the rings of polynomails in many variabels. We briefly mentioned symmetric polynomials. pp. 152-155, 159 (Thm. 8.1).

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Lecture on 4/23/04. We summarized the standard linear algebra concepts over an arbitrary field (vector spces, generating sets, linearly independent systems, bases). pp. 171-174.

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Lecture on 4/26/04. We talked on bases and dimension. pp. 179-181.

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Lecture on 4/28/04. We talked on matrices and linear maps (i.e. homomorphisms of vector spaces). In particular, the ring of n by n matrices over a field K is naturally isomorphim of the ring of homomorphisms from K^n to K^n. By the same token we completed our survey of linear algebra. In the next lectures these concepts will be generalized to modules over general commutative rings. pp.182-185.

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Lecture on 4/30/04. We introduced modules and considered the following examples: submodules of a ring are the same as ideals of this ring, For an ideal I in a ring R the quotient group R/I is a module over R in a natural way, free modules are defined as those having bases (not all modules are free, e.g. the ideal (X,Y) in K[X,Y], K a field, is not a free module over K[X,Y]). A part of this is covered on pp. 186-186.

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Lecture on 5/03/04. We introduced homomorphisms (in particular, isomorphisms) between modules. Then we considered the followinb example: the ring Z x Z (pairs of integer numbers), I=Z x 0 the ideal consisting of the pairs with second component = 0 and J=0 x Z - the pairs whose fisrt component is 0. It turns out that I and J are isomorphic as abelian groups (obvious) but there does not exist an R-module isomorphism between them. p. 188 (except the mentioned example).

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Lecture on 05/07/04 and 05/10/04. We considred several constructions that produce new modules out of olds ones: direct sums, Ker(f), Im(f), Hom(M,N), M/N. We observed that: (i) abelian groups are the same as Z-modules, (ii) there are no nonzero group homomorphisms from Q to Z, (iii) there are no nonzero group homomorphisms from Z/mZ to Z/nZ when g.c.d.(m.n)=1, (iv) for an R-module M there is a natural isomorphism between Hom_R(R,M) and M, given by the assignment f --> f(1). pp.193,197-198.

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Lecture on 05/14/04 and 05/17/04. We introduced the notion of a cyclic module and stated the claim that over a PID all finitely generated modules are direct sums of cyclic modules. The same statement remains true for arbitrary principal ideal ring, i.e. a not necessarily integral ring whose every ideal principal. pp. 204-205 contain much more stronger results.