这个命题是对的吧？

[TheMatrix]

一个有限元素的integral domain一定是一个field。

[TheMatrix]

一个有限元素的integral domain一定是一个field。

嗯。Bing说这个是对的，还给了证明。也不难理解。

[FoxMe]

这可以解释为啥数域中，prime ideal 也是maximal ideal。

[FoxMe]

还又个推广：Wedderburn's little theorem states that every finite division ring is a field. In other words, for finite rings, there is no distinction between integral domains, division rings and fields.

[TheMatrix]

这可以解释为啥数域中，prime ideal 也是maximal ideal。

哦对啊。因为数域的ideal的index，也叫ideal norm吧，都是有限。也就是quotient ring为有限。所以prime ideal的quotient为有限的integral domain，所以必然是一个有限域。所以prime ideal也必然是maximal。

[TheMatrix]

还又个推广：Wedderburn's little theorem states that every finite division ring is a field. In other words, for finite rings, there is no distinction between integral domains, division rings and fields.

谢谢。