Ring-theoretic constructions

Show that the ring $Q_{M}(\mathbb {Z})$ of all rational numbers $a / b$ with $b$ not divisible by a fixed prime number $p$ contains a unique maximal ideal

Any ring possessing a unique maximal ideal is called a local ring.

Show that in any local ring $K$ with maximal ideal $\mathfrak {m}$, the elements not lying in $\mathfrak {m}$ are invertible.

An ideal $\mathfrak {p}$ of a ring $K$ with identity is called prime if the quotient ring $K / \mathfrak {p}$ is an integral domain. Every maximal ideal is prime. The complement $M=K \backslash \mathfrak {p}$ in the ring $K$ is a multiplicative subset (a monoid not containing 0). Under these conditions the ring $Q_{M}(K)$ is more often denoted by the symbol $M^{-1} K$ or simply $K_{\mathfrak {p}}$.

Show that the ring $K_{\mathfrak {p}}$ is always local and that its maximal ideal $\mathfrak {m}_{\mathfrak {p}}$ consists of quotients of the form $a / b$, where $a \in \mathfrak {p}, b \in K \backslash \mathfrak {p}$. Show also that $\mathfrak {m}_{\mathfrak {p}} \cap K=\mathfrak {p}$.

The operation of passing from $K$ to the local ring $K_{\mathfrak {p}}$ is called localization of the ring $K$ with respect to the prime ideal $\mathfrak {p}$.