# Commutative Rings, Ideals, Kernels, Matrices and Injective and Surjective Ring Homomorphisms

If n Є R and R is a commutative ring we indicate by Mn(R) the ring of allnxn entries wrt the usual operations on matrices. If n>1 this ring is commutative even if R is.

Let S={(aij)ЄMn(R)|i≠j=>aij=0}

Let k be an integer 1≤k≤n. Show that

a) S is a commutative subring of Mn(R)

b) The function f: S-->R defined by f((aij))=akk is a surjective ring homomorphism

c) The set defined by IK={(aij) Є S | akk=0} is the kernel of f

d) IK is an ideal of S

What are necessary conditions for S to be an integral domain?

e) Show that R ={[a 0]|a,b,c Є R} is a subring of M2(R). Is it commutative? Find a non trivial ideal of R.

[b c]

f) Is S ={[a b]|a,b,c Є R} is a subring of M2(R)?

[c 0]

g) Show that the function C-->M2(R) a+bi -->[a b] is an interjecting homomorphism.

[-b a]

Please see the attached file for the fully formatted problems.

© BrainMass Inc. brainmass.com October 9, 2019, 5:23 pm ad1c9bdddfhttps://brainmass.com/math/ring-theory/commutative-rings-ideals-kernels-matrices-and-injective-and-surjective-ring-homomorphisms-55911

#### Solution Summary

Commutative Rings, Ideals, Kernels, Matrices and Injective and Surjective Ring Homomorphisms are investigated. The solution is detailed and well presented.