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:
- $(R,\oplus)$ is an abelian group.
- Associativity of multiplication holds: $a\cdot(b\cdot c) = (a\cdot b)\cdot c$.
- 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?