Let R be a commutative ring. Prove that Hom_R(R, M) and M are isomorphic R-modules

alesterp 2021-01-19 Answered
Let R be a commutative ring. Prove that HomR(R,M) and M are isomorphic R-modules
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Expert Answer

opsadnojD
Answered 2021-01-20 Author has 95 answers
Define F:Hom(R,M)M by F(ϕ)=ϕ(1)M
First, we proved that F is an R-module homomorphism.
Let ϕ,ρHom(R,M)andrR. Then,
F(rϕρ)=(rϕρ)(1)=rϕ(1)ρ(1)=rF(r)F(r)
Hence, F is an R-module homomorphism.
Now, to prove that it is an isomorphic R-module, we must prove that F is bijective.
Suppose F(ϕ)=0 for some ϕHom(R,M). Then, ϕ(1)=0.
For any rR,ϕ(r)=rϕ(1)=r0=0. Hence, ϕ=0. Thus, F is injective.
Now, suppose mM.
Define a map ϕ:RM by ϕ(r)=rm for any rR.
Let r,s,tR.
ϕ(rst)=(rst)m=r(sm)tm=rϕ(s)ϕ(t)
Hence, ϕHom(R,M).
Futher, F (ϕ)=ϕ(1)=m.
Thus, F is surjective.
F is homomorphism and bijective. Therefore, Hom(R,M)andM are isomorphic R-module.
Not exactly what you’re looking for?
Ask My Question

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Relevant Questions

asked 2020-12-09
If A and B are ideals of a commutative ring R with unity and A+B=R show that AB=AB
asked 2022-01-04
Solve following system of equations of non-negative integers using methods of Commutative Algebra
3σ1+2σ2+σ3+2σ4=10
4σ1+3σ2+σ3=12
2σ1+4σ2+2σ3+σ4=25
asked 2020-10-19
Show that a ring is commutative if it has the property that ab = ca implies b = c when a0.
asked 2022-01-07
Show that L1(R) a Banach algebra Commutative
asked 2020-11-08
Suppose that a and b belong to a commutative ring and ab is a zero-divisor. Show that either a or b is a zero-divisor.
asked 2022-05-20
Classifying all commutative R-algebras of matrices over R?
I initially thought they were all isomorphic to some subring of the n × n diagonal matrices D R × × R, but this was wrong: Every commutative ring of matrices over R is isomorphic to the diagonals?. One counterexample is matrices of the form (using block matrix notation) [ α I 1 A 0 α I n 1 ] for some 1 × ( n 1 ) real matrix block A and some α R, which forms a commutative ring ( U , + , ).
Are there other counterexamples? Can we classify all such rings up to isomorphism?
asked 2022-05-24
Let g be a Lie algebra and let a , b , c g be such that a b = b a and [ a , b ] = c 0. Let h = s p a n   { a , b , c }. How to prove that h is isomorphic to the strictly upper triangular algebra n ( 3 , F )?
Problem: If h n ( 3 , F ) then a , b , c n ( 3 , F ) with a b = b a and [ a , b ] = c as in h But then c must equal 0 whereas c h is not 0?

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question