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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**Example text**

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 .