What is the difference between $7\mathbb{Z}$ and (7)?

Question

For example in the context of prime ideals, the prime ideal for the integers are the sets that contain all the multiples of a given prime number, so would a prime ideal for the ring $\mathbb{Z}$ be $7\mathbb{Z}$ or (7) ? And what is the difference between the two?

Thanks

Answer 1

For a commutative ring $R$ we write $I=(a)=aR$ for the principal ideal generated by $a$. Similarly, for the quotient ring, one can write $R/(a)$ or $R/aR$. For $R=\Bbb Z$ and $a=7$ this would be $\Bbb Z/7\Bbb Z$ or $\Bbb Z/(7)$. Some people even write $\Bbb Z_7$, but this is in conflict with the ring of $7$-adic integers.