An Introduction to Linear Analysis by Donald L. Kreider, Robert G. Kuller, Donald R. Ostberg,

By Donald L. Kreider, Robert G. Kuller, Donald R. Ostberg, Fred W. Perkins, Lynn H. LOoomis

An creation to Linear research

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Given a ring R and a positive integer n, we use Mn (R) to denote the ring of all n × n matrices over R. The standard n × n matrix units in Mn (R) are the matrices eij (for i, j = 1, . . , n) such that eij has 1 for the i, j-entry and 0 for all other entries. 7. Let R be a right noetherian ring and S a subring of a matrix ring Mn (R). If S contains the subring 0 ··· 0  0 r ··· 0 r R = . . ..  r ∈ R .. 0 0 ··· r of all “scalar matrices,” then S is right noetherian. In particular, Mn (R) is a right noetherian ring.

Observe that the diagonal subring S0 T0 is isomorphic to S × T and so is right noetherian. Observe also that if elements b1 , . . , 0 bn 0 0 generate R as a right S0 T0 -module. 5 shows that R is right noetherian. Conversely, assume that R is right noetherian. Observing that the projection maps 0s bt → s and 0s bt → t are ring homomorphisms of R onto S and of R onto T , we see that S and T must be right noetherian. , 0 bn 0 0 , from which we infer that the elements b1 , . . , bn generate BT .

A) Show that formal matrix addition and multiplication make sense in , and that by using those operations S0 B becomes a ring. T S B 0 T (b) Show that there is also a ring and that T 0 B S ∼ = S B 0 T T 0 B S of formal lower triangular matrices, . (c) Observe that the set 0B 0 0 of matrices 0 b 00 is an ideal of and that, under the obvious abelian group isomorphism of B onto 0B 0 0 S B 0 T , , left S-submodules (right T -submodules, (S, T )-sub-bimodules) of B correspond precisely to left ideals (right ideals, two-sided ideals) of S0 B contained in T 0B 0 0 .

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