Question: Problem 6. Suppose that R = ( R , + , * ) is a commutative ring , and let'S be a proper ideal of

Problem 6. Suppose that R = ( R , + , * ) is a commutative ring , and let'S be a proper ideal of R. Let a E R be such that a ES and let Sa = [ m + a * 6 : ME S and b ER ) Part ( a ) . Prove that Sa is an ideal of R . Part ( b ) . If's is a maximal ideal of R, explain why we must have Sa = R. Part ( C ) . If'S is a maximal ideal of & and * ER, explain why there must exist Y ER with the property that x - a * YES. ( Consequently, a* V E x + 5. )

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