Module theory Notes, Advanced mathematics, Abstract Algebra notes, Handwritten notes

Module definition

 

Module Example

Module Example

Z_module 

Z_module and the difference between vector space and module

sub-module criteria

example of sub-module

examples of sub-module

sub-module examples solution

sub-module examples solution

sub-module example solution

is R^n be sub-module?

Rn be sub-module

module homomorphism

every ring homomorphism is module homomorphism but the converse is not true.

a map is an R-module homomorphism

an abelian group is a ring homomorphism

elements of ring homomorphism are module homomorphism

ring homomorphism is group

ring homomorphism is abelian

endomorphism, quotient module, R-module is homomorphism with kernel

Quotient group under the action of ring elements

projection map is a module homomorphism

sub-module, the sum of the module,1st isomorphism theorem of the module

2nd,3rd,4th isomorphism theorems of module

 proposition

pi is surjective

cyclic module

finitely generated module

R module is finitely generated 

direct product, free module

free module example, direct product is R module 

Module fundamentals

Ring theory Notes, Mathematics, Handwritten notes

notations of ring

ring definition

difference between group and ring

 

null set form a smallest ring

commutative ring

ring theory

ring with unity

polynomial ring

division ring or skew field

cancellation law hold in Integral domain

integral domain hold in cancellation law

Field

A field is integral domain

Integral domain is not field

Ring homomorphism

counter example of ring homomorphism

ring isomorphism

example of ring isomorphism

Ideal definition, ideal contain a unit

ideal in R are <0> and itself

types of ideal, principal ideal, prime ideal

Examples and counter examples of ideal

Maximal ideal

lemma and theorem related to ideals

Quotient ring or factor ring

example of quotient ring and ideal

characteristic of a ring ,finite field

characteristic of field is zero or a prime number

relationship between commutative ring and field

 R/M is field

M is maximal ideal 

kernal f is ideal

1st theorem of isomorphism

first theorem of isomorphism part 2

2nd theorem of isomorphism

2nd theorem isomorphism part 2

2nd theorem of isomorphism part 3

fundamental theorem of ring homomorphism

fundamental theorem of isomorphim part 2

ring Z is trivial isomorphism with itself

Z is trivial isomorphism with itself part 2

zero divisor 

Ring theory, Pure mathematics, Abstract Algebra

 Ring theory is commonly seen as a subject in Pure Mathematics. This implies it is a subject of natural magnificence. In any case, the possibility of a ring is fundamental to the point that it is additionally crucial in numerous utilizations of Mathematics. Without a doubt it is fundamental to the point that a lot of other essential apparatuses of Applied Mathematics are worked from it. For example, the vital idea of linearity, and straight algebra, which is a down to earth need in Physics, Chemistry, Biology, Finance, Economics, Engineering, etc, is based on the thought of a vector space, which is a unique sort of ring module. Ring theory seems to have been among the most loved subjects of the absolute most compelling Scientists of the twentieth century, for example, Emmy Noether; and Alfred Goldie. In any case, maybe more essential than any of these focuses is that ring theory is a center piece of the subject of Algebra, which frames the language inside which present day Science can be put on its firmest conceivable balance.