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By Harold F. Schiffman

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1 are satisfied. The submodules M1 and M2 are called direct summands of the module M . The internal direct sum of several modules can be defined in a similar way. For this purpose we shall prove the following statement. 2. Let Mi (i ∈ I) be a family of submodules of a module M , and f : ⊕ Mi → M be the homomorphism defined by the formula f (⊕i mi ) = i mi . i∈I Then the following conditions are equivalent: 1) f is an isomorphism; 2) Mi = M and Mi ∩ ( Mj ) = 0 for any i; i∈I j=i Mi = M and Mi ∩ ( 3) i∈I Mj ) = 0 for any i > 1.

Suppose f = 0. Since α(f ) = i∈I fi ai = 0 if and only if all ai = 0. Hence, any element f ∈ F can be f = i∈I fi ai with ai ∈ A. Such a set of elements uniquely written as a finite sum i∈I { fi ∈ F : i ∈ I} is called a free basis for F . Conversely, let a module F have a free basis { fi ∈ F : i ∈ I}. Then ALGEBRAS, RINGS AND MODULES 26 fi ai = 0 if and only if all ai = 0. Therefore a map ⊕ A → F given by f = i∈I i∈I ai → i fi ai is an isomorphism. i Hence, we obtain the following result. 3. A module F is free if and only if it has a free basis.

0), where the identity of the ring Ai is at the i-th position and zeroes elsewhere. , en are pairwise orthogonal idempotents and e1 + e2 + ... + en is the identity of A. , the idempotents ei are in the center of the ring A. Such idempotents are said to be central. , n, then we denote the direct product by An = A × A × ... × A. , n) A = Ai . , 0) ∈ A, where ai ∈ Ai , forms an ideal Ii in A. Then the ring A, considered as the regular module, is a direct sum of the ideals Ii . Conversely, let A = I1 ⊕ ...

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