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