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Pespectives in Mathematical Sciences Due: Tuesday, June 16, 2020, on NUCT. Problem 1. Let R be a ring, and let R

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Pespectives in Mathematical Sciences Due: Tuesday, June 16, 2020, on NUCT.

Problem 1. Let R be a ring, and let Rop be the opposite ring defined in Re- mark 2.6. LetMn(R) be the ring ofn×n-matrices with entries inR, and let

(−)t:Mn(R)op→Mn(Rop)

be the map that to a matrixA= (aij) assigns its transposeAt= (aji).

(a) Show that (−)tis a ring homomorphism.

(b) Show that (−)tis a ring isomorphism.

Problem 2. LetDbe a division ring, and letR=Mn(D) be the matrix ring. The set S =Mn,1(D) of column vectors has both a structure of leftR-module and of rightD-module with sum given by matrix sum and scalar multiplication given by matrix product. Moreover, for allA∈R,x∈S, anda∈D, (A·x)·a=A·(x·a), by the associativity of matrix product. Show that the map

Dop ρ //EndR(S) defined byρ(a)(x) =x·ais a ring isomorphism.

1

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