Given a finite ring, e.g $\mathbb{Z}{_{24}}$, how can I find all of its subrings?
I have tried to think about it couldn't reach any idea.
Thanks.
2026-02-23 12:04:18.1771848258
Find all subrings of a ring
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A "brute force" way of doing this is by constructing all the subrings generated by one element, by two elements, and so forth until you have enough generators to yield the entire (finite) ring you started with.
Of course we can be a bit clever about how we do this. E.g. once you have all the one-element generated subrings, you only want to consider two-element generated subrings where the second generator does not belong to the subring generated by the first element.
In the specific case of $\mathbb{Z}/24\mathbb{Z}$, assuming you require the identity element, every subring must contain the subring generated by $1$. This happens to be all of $\mathbb{Z}/24\mathbb{Z}$, and we would be done. This will not be the general case however, and if you wanted to consider (sub)rings without unity, you'd be looking at ideals and some other subsets of $\mathbb{Z}/24\mathbb{Z}$.