Commutative Rings, Ideals, Kernels, Matrices and Injective and Surjective Ring Homomorphisms
Not what you're looking for?
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.
Purchase this Solution
Solution Summary
Commutative Rings, Ideals, Kernels, Matrices and Injective and Surjective Ring Homomorphisms are investigated. The solution is detailed and well presented.
Purchase this Solution
Free BrainMass Quizzes
Know Your Linear Equations
Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.
Probability Quiz
Some questions on probability
Solving quadratic inequalities
This quiz test you on how well you are familiar with solving quadratic inequalities.
Graphs and Functions
This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.
Geometry - Real Life Application Problems
Understanding of how geometry applies to in real-world contexts