Transcription of Ring Theory Problem Set 1 { Solutions be a ring with unity ...
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Ring Theory Problem Set 1 { Solutions be a ring with unity ...
ring Theory Problem Set 1 SolutionsProblem a ring with unity 1. Show that ( 1)a= afor alla :We have 1 + ( 1) = 0 by definition. Multiplying that equation on the rightbya, we obtain(1 + ( 1)) a= 0 a= 0by theorem , part i. By the distributive law, we obtain the equation1 a+ ( 1) a= 0and therefore we havea+( 1)a= 0. We also havea+( a) = 0. Thus,a+( 1)a=a+( a).The ringRunder addition is a group. The cancellation law in that group implies that a= ( 1)awhich is the result we wanted to a field and leta, b F. Assume thata6= 0, Show that thereexists an elementx Fsatisfying the equationax+b= :SinceFis a field anda6= 0, there exists an elementa 1inFsuch thataa 1= 1.
Ring Theory Problem Set 1 { Solutions Problem 16.1 Let Rbe a ring with unity 1. Show that ( 1)a= afor all a2R. SOLUTION: We have 1+( 1) = 0 by de nition. Multiplying that equation on the right by a, we obtain 1 + ( 1) a = 0 a = 0 by theorem 16.1, part i. By the distributive law, we obtain the equation 1 a+ ( 1) a = 0 and therefore we have a+( 1 ...
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