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(1) Every module M has trivially M itself as a generating set (for every me M is a finite linear combination of the form m = m1, 1€ R). (2) If R is a ring, then {1} is a basis of RR (and of RR). (3) We now consider properties of Qz. If finitely many arbitrary elements are omitted from an arbitrary generating set X of Qz, then the set with these elements omitted is again a generating set of Qz

(1) Every module M has trivially M itself as a generating set (for every me M is a finite linear combination of the form m = m1, 1€ R). (2) If R is a ring, then {1} is a basis of RR (and of RR). (3) We now consider properties of Qz. If finitely many arbitrary elements are omitted from an arbitrary generating set X of Qz, then the set with these elements omitted is again a generating set of Qz

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter5: Rings, Integral Domains, And Fields
Section5.2: Integral Domains And Fields
Problem 5E: Examples 5 and 6 of Section 5.1 showed that P(U) is a commutative ring with unity. In Exercises 4...
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(1) Every module M has trivially M itself as a generating set (for every me M is a finite linear combination of the form m = m1, 1€ R). (2) If R is a ring, then {1} is a basis of RR (and of RR). (3) We now consider properties of Qz. If finitely many arbitrary elements are omitted from an arbitrary generating set X of Qz, then the set with these elements omitted is again a generating set of Qz
Transcribed Image Text:(1) Every module M has trivially M itself as a generating set (for every me M is a finite linear combination of the form m = m1, 1€ R). (2) If R is a ring, then {1} is a basis of RR (and of RR). (3) We now consider properties of Qz. If finitely many arbitrary elements are omitted from an arbitrary generating set X of Qz, then the set with these elements omitted is again a generating set of Qz
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