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).  

No comments:

PDF FILE TO YOUR EMAIL IMMEDIATELY PURCHASE NOTES & PAPER SOLUTION. @ Rs. 50/- each (GST extra)

HINDI ENTIRE PAPER SOLUTION

MARATHI PAPER SOLUTION

SSC MATHS I PAPER SOLUTION

SSC MATHS II PAPER SOLUTION

SSC SCIENCE I PAPER SOLUTION

SSC SCIENCE II PAPER SOLUTION

SSC ENGLISH PAPER SOLUTION

SSC & HSC ENGLISH WRITING SKILL

HSC ACCOUNTS NOTES

HSC OCM NOTES

HSC ECONOMICS NOTES

HSC SECRETARIAL PRACTICE NOTES

2019 Board Paper Solution

HSC ENGLISH SET A 2019 21st February, 2019

HSC ENGLISH SET B 2019 21st February, 2019

HSC ENGLISH SET C 2019 21st February, 2019

HSC ENGLISH SET D 2019 21st February, 2019

SECRETARIAL PRACTICE (S.P) 2019 25th February, 2019

HSC XII PHYSICS 2019 25th February, 2019

CHEMISTRY XII HSC SOLUTION 27th, February, 2019

OCM PAPER SOLUTION 2019 27th, February, 2019

HSC MATHS PAPER SOLUTION COMMERCE, 2nd March, 2019

HSC MATHS PAPER SOLUTION SCIENCE 2nd, March, 2019

SSC ENGLISH STD 10 5TH MARCH, 2019.

HSC XII ACCOUNTS 2019 6th March, 2019

HSC XII BIOLOGY 2019 6TH March, 2019

HSC XII ECONOMICS 9Th March 2019

SSC Maths I March 2019 Solution 10th Standard11th, March, 2019

SSC MATHS II MARCH 2019 SOLUTION 10TH STD.13th March, 2019

SSC SCIENCE I MARCH 2019 SOLUTION 10TH STD. 15th March, 2019.

SSC SCIENCE II MARCH 2019 SOLUTION 10TH STD. 18th March, 2019.

SSC SOCIAL SCIENCE I MARCH 2019 SOLUTION20th March, 2019

SSC SOCIAL SCIENCE II MARCH 2019 SOLUTION, 22nd March, 2019

XII CBSE - BOARD - MARCH - 2019 ENGLISH - QP + SOLUTIONS, 2nd March, 2019


BUY FROM PLAY STORE

DOWNLOAD OUR APP


HOW TO PURCHASE OUR NOTES?



S.P. Important Questions For Board Exam 2020


O.C.M. Important Questions for Board Exam. 2020


Economics Important Questions for Board Exam 2020


Chemistry Important Question Bank for board exam 2020


Physics – Section I- Important Question Bank for Maharashtra Board HSC Examination


Physics – Section II – Science- Important Question Bank for Maharashtra Board HSC 2020 Examination


MUST REMEMBER THINGS on the day of Exam


Are you prepared? for English Grammar in Board Exam.


Paper Presentation In Board Exam


How to Score Good Marks in SSC Board Exams


Tips To Score More Than 90% Marks In 12th Board Exam


How to write English exams?


How to prepare for board exam when less time is left


How to memorise what you learn for board exam


No. 1 Simple Hack, you can try out, in preparing for Board Exam

How to Study for CBSE Class 10 Board Exams Subject Wise Tips?

JEE Main 2020 Registration Process – Exam Pattern & Important Dates

NEET UG 2020 Registration Process Exam Pattern & Important Dates

How can One Prepare for two Competitive Exams at the same time?

8 Proven Tips to Handle Anxiety before Exams!