Question #101847
Prove or disprove:
a) every left simple ring is right simple
b) every simple ring is left simple
1
Expert's answer
2020-01-29T13:54:22-0500

a) A ring RR with 1 is called simple if (0)(0) and RR are the only two-sided ideals of RR .

If RR is a Artinian ring then left simple ring is right simple. By the Artin-Wedderburn theorem, a ring RR is simple and (left or right) Artinian if and only if R=Mn(R)R=M_{n}(R) ( Mn(R)M_{n}(R) be the ring of n×nn\times n matrices with entries from RR ) for some division ring RR and some integer n1n\geq1 .


RR every ring that the statement is incorrect because

subject to the problem RR every left simple ring

iI,rR,riI\forall i\in I, \forall r\in R, ri\in I where II ideals of RR , II is not (0)(0) and RR

ir∉Iir\not \in I in the general case.




b) Every simple ring is left simple is correct. A ring RR with 1 is called simple if (0) and RR are the only two-sided ideals of RR .


Subject to the problem RR every simple ring

iI,rR,riI,irI\forall i\in I, \forall r\in R, ri\in I, ir\in I where II ideals of RR , II is not (0)(0) and RR . Whereas riIri\in I RR

is left simple ring


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