Which of the following is not necessarily true in a ring with identity R? 1) For

Question

Which of the following is not necessarily true in a ring with identity R? 1) For all a, b, c in R, (a · b) · c = a · (b · c) 2) There is a multiplicative identity (meaning there is an element 1 such that 1 · a = a · 1 = a) 3) For all a and b in R, a + b = b + a. 4) For all a and b in R, a·b = b·a 5) Every element a has an additive inverse (an element b such that a + b = b+ a = 0) 6) There is an additive identity (meaning there is an element 0 such that 0 + a = a + 0 = a). 7) R is closed under the operations + and · 8) For all a, b, c in R, a · (b + c) = a · b + a · c 9) For all a, b, c in R, (a + b) + c = a + (b + c)

Explanation / Answer

A ring is a non-empty set R together with a two binary operations + and . satisfying the properties:

On the basis of above description, it is not necessary that it follows

2) There is a multiplicative identity (meaning there is an element 1 such that 1 · a = a · 1 = a)

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