Question

On the set of integers Q, define the operations a⊕b:=a+b+1 and a⊙b:=a+b+ab Note that R=(Q,⊕,⊙) is a commutative ring (Do not prove that!). Find the zero element.

Answer 1

To find the zero element for the operation a⊕b := a + b + 1 in the given ring, we set a ⊕ z = a. Solving for z, we determine that z = -1 is the zero element of the ring, as it leaves any element a unchanged under the operation ⊕.

Step-by-step explanation:

The zero element in a ring is the additive identity, which means for any element a in the ring, a ⊕ zero element = a. Given the operation a⊕b := a + b + 1, we want to find an integer z such that for any integer a, a ⊕ z = a. Substituting the operation we have:

• a ⊕ z = a + z + 1 = a

This means z + 1 must be 0, so z = -1 is the zero element for the ⊕ operation in the given ring.