Commutative Algebra


Introduction
      Commutative Algebra is essentially the study of commutative rings. One of the things which distinguishes the modern approach to Commutative Algebra is the greater emphasis on modules, rather then just on ideals. The notion of an ideal which arose from number theory is also important in Algebraic Geometry, It is useful to study ideals from a module theoretic set up, as operations of linear algebra such as formation of quotients, products and tenser products are closed for modules but not for ideals.
          This project is concerned with a preliminary study of modules, The second chapter deals with modules, sub modules and homomorphism of modules. The third chapter introduces the free modules and finitely generated modules. The fourth chapter deals with projective modules and their elementary properties. Projective modules play the role of a vector space while studying linear algebra over a general commutative ring.
The last chapter introduces the tensor product of modules. Sometimes geometric objects defined over a field behave differently over a bigger field. Such questions are best studied by scalar extension of ring tensor products. It also deals with the existence and uniqueness of the tensor product.


                                           


   


                                             Chapter-I
                                                         Preliminary
Definition : 1.1
         A nonempty set of elements G is said to form  a group if G there is defined a binary operation, called the product and denoted by  such that
i.                 a ,b ЄG implies that a.b ЄG (closed).
ii.               a,b,c ЄG implies that a.(b.c) = (a.b).c (associative law).
iii.             there exists am element e Є G such that a.e = e.a =a for all
a Є G(the existence of an identity element in G)
iv.             for every a Є G there exist an element Є G such that a. = .a=e (the existence of inverses in G).
Definition 1.2
       A group G is said to be abelian if for every a,b Є G, a.b =b.a.
Definition 1.3
       A non empty subset H of a group G is said to be a subgroup of G if under the product in G, itself forms a group.
Definition 1.4
   A non empty set R is said to be an association ring if in R there are defined two operation, denoted by + and . respectively, such that for all a,b,c in R:
1.    a+b is in R
2.    a+b = b+a
3.    (a+b) =c = a +(b+c)
4.    there is an element 0 in R such that a+0=a (for every a in R).
5.    there exists an element –a in Rsuch that a =(-a)=0.
6.    a.b is in R.
7.    a.(b.c) = (a.b).c
8.    a.(b+c)= a.b+a.c and(b+c).a = b.a+c.a(the two distributive laws)
If  the multiplication  of R is such that a.b =b.a for every a,b in R,then we
call R, a commutative ring.
Definition 1.5
         A non empty subset U of R is said to be ideal of R if
1.    U is a subgroup of R under addition
2.    for every u ЄU and r ЄR ,both ur and ru are in U.
Definition   1.6
       An ideal M  R  in a ring R is said to be a maximal ideal of R if when ever U is an ideal of R such that M  U R, then either R=U or M=U.


Definition 1.7
      A non-empty set V is said to be a vector space over a field F if V is an abelian group under an operation which we denote by +, and if for every
αЄF, vЄ V there is defined an element, written α v , in V subject to
1.    α(v + w) = αv + α w;
2.    (α + β) v=  α v +β v;
3.    α(βv) =(αβ)v;
4.    1 v=v
for all  α, β Є F ,v ,w Є V (where the 1 represents the unit element of F under multiplication).