Let A be a commutative ring, and BCA be an ideal. By definition, know an annihilator...

 

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Let A be a commutative ring, and BCA be an ideal. By definition, know an annihilator of a subset S of a module over a ring is the ideal formed by the elements of the ring that give always zero when multiplied by an element of S. Define the annihilator of B to be be the subset of elements that multiply to 0 with any element of B such that Ann(B) = {a € A: Vb € B, ab=0} Prove that Ann(B) is an ideal. Show all work. Let A be a commutative ring, and BCA be an ideal. By definition, know an annihilator of a subset S of a module over a ring is the ideal formed by the elements of the ring that give always zero when multiplied by an element of S. Define the annihilator of B to be be the subset of elements that multiply to 0 with any element of B such that Ann(B) = {a € A: Vb € B, ab=0} Prove that Ann(B) is an ideal. Show all work.