# Finiteness of the fibers of the prime spectrum [duplicate]

Let $A\rightarrow B$ be a ring homomorphism such that $B$ is a finitely generated $A$-module. How one shows that the (set-theoretic) fibers of the map $\operatorname{Spec}B\rightarrow\operatorname{Spec}A$, where the spectra are considered as topological spaces, are always finite?
## marked as duplicate by user26857 abstract-algebra StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Dec 12 '16 at 20:20
• This is related to the topic of integral extensions of rings, just in a different language. IIRC, just rephrase the problem in terms of prime ideals of $B$ lying over prime ideals of $A$. – user14972 Dec 12 '16 at 20:16