# Linear Associative Algebras by Alexander Abian (Auth.)

By Alexander Abian (Auth.)

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Nt are natural numbers r l9 r2, . . , rt elements of & and Gu G 2 , . . , Gt elements of ^. Prove that a left module Jt over a ring satisfies the Ascending Chain Condition (see Problem 3) if and only if every submodule of Jt is finitely generated. 9. Define the notions of the Descending and Ascending Chain Conditions for a ring (by replacing the notions of a submodule by that of a left ideal in the corre­ sponding definitions for a left module given in Problems 2 and 3). Prove that if a ring 3\$ satisfies the Ascending (Descending) Chain Condition for left ideals then any finitely generated unitary left module (see page 40) over 3% satisfies the Ascending (Descending) Chain Condition.

If % is a subspace of a finite dimensional vector space V then d i m ^ ^ dimT (23) LEMMA 4. Let T be an n > 0 dimensional vector space. Then every linearly independent subset Sf ofV with n elements is a basis ofV. Proof Clearly, [Sf\ C T. On the other hand if [Sf] 4= T then by Theorem 6 we have n = dim^ 7 < d i m ^ = n which is a contradiction. Thus, \&\ = T and Sf is a basis of r . 5. Ler V be ann dimensional vector space. Then every subset with m > n elements is linearly dependent. LEMMA SfofV Proof.

Determine the number of elements of an m dimensional vector space over §f. 8. Let T be an m dimensional vector space and 8P and 31 be subspaces of T respectively of dimensions p and q. Prove that if p + q > m then @ and J have a nonzero vector in common. 9. Prove (39) on page 55. 10. Let (fly) be the matrix corresponding to the endomorphism ^> and the ordered basis {Bu B2,... ,Bn} of a vector space 2^. Let {Vt, V2,... , ofr. 3. , an element of \$) by a matrix. These operations are given by: and (atj) 4- (bij) = (c«) with j(fl«) = U % ) .

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