Exercises in Classical Ring TheorySpringer-Verlag, 1995 - 287 pages |
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... unit } is a group under multiplication , and is called the group of units of R. If R { 0 } and U ( R ) = R \ { 0 } ... unit , then a , b are units " ? Show the following for any ring R : ( a ) If a ” is a unit in R , then a is a unit in R ...
... unit } is a group under multiplication , and is called the group of units of R. If R { 0 } and U ( R ) = R \ { 0 } ... unit , then a , b are units " ? Show the following for any ring R : ( a ) If a ” is a unit in R , then a is a unit in R ...
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... unit times an idempotent . ( 2 ′ ) Every a Є R can be written as an idempotent times a unit . If R satisfies ( 1 ) , it is said to be unit - regular . ( 3 ) Show that any unit - regular ring R is Dedekind - finite . Solution . By left ...
... unit times an idempotent . ( 2 ′ ) Every a Є R can be written as an idempotent times a unit . If R satisfies ( 1 ) , it is said to be unit - regular . ( 3 ) Show that any unit - regular ring R is Dedekind - finite . Solution . By left ...
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... unit - regular . Suppose M = KONK ' → N ' , where N N ' . Define a R such that a ( K ) = 0 and a N is a fixed isomorphism from N to N ' . Write a = aua , where u € U ( R ) . As in Exercise 14A , ( * ) M = ker ( a ) im ( ua ) = K → u ...
... unit - regular . Suppose M = KONK ' → N ' , where N N ' . Define a R such that a ( K ) = 0 and a N is a fixed isomorphism from N to N ' . Write a = aua , where u € U ( R ) . As in Exercise 14A , ( * ) M = ker ( a ) im ( ua ) = K → u ...
Table des matières
2 Semisimplicity | 16 |
Jacobson Radical Theory | 35 |
5 Jacobson radical under change of rings | 52 |
Droits d'auteur | |
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a₁ abelian artinian ring assume automorphism B₁ central idempotents char commutative ring conjugate consider constructed contradiction decomposition Dedekind-finite defined division ring domain element endomorphism equation Exercise exists fact finite group finite-dimensional follows group G hopfian idempotent identity implies indecomposable integer inverse irreducible isomorphism J-semisimple Jacobson radical k-algebra kG-module left ideal left primitive ring Lemma linear local ring M₁ matrix maximal ideal maximal left ideal maximal subfield minimal left Mn(R module multiplication Neumann regular ring nil ideal nilpotent ideal noetherian ring noncommutative nonzero polynomial prime ideal primitive idempotents primitive rings proof prove quasi-regular R-module R/rad rad kG representation resp right ideal right R-module ring theory semilocal ring semiprime semisimple ring show that rad simple left R-module simple ring soc(RR Solution stable range strongly regular subdirect product subgroup subring Theorem unit-regular zero