Commutative Ring with Identity

Question

How can I show that $(\mathbb Q,\oplus,\cdot)$ is a commutative ring with identity where $\oplus$ and $\cdot$ are defined as, $a\oplus b=a+b-1$ and $a\cdot b=a+b$?

According to the book, an algebraic structure $(R,\oplus,\cdot)$ is called a ring if the following conditions are satisfied:

1. $(R,\oplus)$ is an abelian group.
2. Associativity of multiplication holds: $a\cdot(b\cdot c) = (a\cdot b)\cdot c$.
3. The left distributive law $a\cdot(b\oplus c)=(a\cdot b)\oplus(a\cdot c)$ and the right distributive law $(b\oplus c)\cdot a=(b\cdot a)\oplus(c\cdot a)$ are satisfied by "$\oplus$" and "$\cdot$".

Though I was somehow able to prove first 2 conditions, the third condition is not getting satisfied. It's an "show that..." question, so the statement is definitely true. Can someone help me?

Answer 1

It is obviously not a ring. In what follows, $0$ and $1$ refer to the neutral elements of the original ring addition and multiplication, respectively.

Just note that $1$ is the neutral element for $\oplus$, and so it should satisfy $a\cdot 1=1$ for all $a\in R$, if $R$ is to be a ring. (This is because we know the additive identity is multiplicatively absorbing in a ring.)

But it does not: $a\cdot 1:=a+1\neq 1$, for any $a\neq 0$.

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In case you have radically mistyped your problem, I would encourage you to search for duplicates before asking. This and this and this are all similar, and may explain the answer to you faster than re-asking.