Question

5. An elementa of a ring Ris regular in the sense of Von Neumann) if there exists ER such that axa - a. If every element of R is regular, then R is said to be a regular ring. 3. SEMISIMPLE RINGS (a) Every division ring is regular. (b) A finite direct product of regular rings is regular. (c) Every regular ring is semisimple. The converse is false (for example, Z). (d) The ring of all lincar transformations on a vector space (not necessarily finite dimensional) over a division ring is regular. (e) A semisimple left Artinian ring is regular. (1) R is regular if and only if every principal left (resp. right) ideal of R is generated by an idempotent element. (8) A nonzero regular ring Rwith identity is a division ring if and only if its only idempotents are 0 and IR.

Answer 1

Given that - Ош шох a) Every division ding is regular. Let us, R be a division ring to show it is regular. That is we have to show for any QER < 2 #0 a'er So that, consider #za Then, we get Qoca = aa'a = e.a = a : ano If any then for any XER we have 0.8.0 -0.

finite direct product of regular rings is b) A regular fet us, R, Ra, R3, Ru, RS, ---Ro be regular ring to Show © Ri is also regular for this a= [0,9a, Ag ----,an) E ORI Then, Each ai E Ri i=1, ton Since, Riis Regular do I Hi Eri Sit Qi XpQ- Q Hi - 0 So, 8 = (0,29, 83, du- --80) Then. han! se ki Arte that

ada = (0,12g, a3, --- An) (0,83,883 ---80) (a1, an) = (a, a,, 9, 82,92, 292393 --- analo an ) = (21, 2d, q ---an) [? by 2] a Hence, for any ae © Ri we have & sit axa sa Ô Ri is regular Let, R be a non- zero - regular sing. Then R is division ding. o le are only idempotents. Let, R is a be division ring & the indempotent.

Then a²=a a²a=0 aca-k) =0 a zo oo a-l = 0 a=0 or a sik Since, R has no non- zero, zero divisions Let O & 'R are only idem potents to show that division ving Ris sufficient to show that has increase. non-zero ret, a +0 ER and Ris regular So F HER Siti Axa - a acaa-1) = 0 a=0 or æQ-1=0

xa-1=0 Since, we have to taken a 40 xa=1 g=a-1 Hence, bis division ding Thank you.