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

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    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.

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    https://brainmass.com/math/ring-theory/commutative-rings-ideals-kernels-matrices-and-injective-and-surjective-ring-homomorphisms-55911

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    Solution Summary

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

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