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In general, can someone please explain how to determine what the ideals of a ring are? I understand that an Ideal is a subset of a ring such that it contains any element in the ring multiplied by the ideal is absorbed in the ideal, the ideal is closed under addition, and it contains the identity. But how do you actually determine what the ideals are given a ring?

Some examples / hints/ suggestions would be very helpful - thank you!

Masha
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  • No, an ideal of a ring does not contain the identity, unless it is equal to the whole ring. Roughly speaking, an ideal is a subset of a ring which is stable addition and by scaler multiplication – somewhat like a subspace of a vector space. The simplest ideals are principal ideals , which consist of all multiples of a given element. – Bernard Apr 25 '19 at 23:45
  • Although this is not a sure recipe for finding all ideals, the following fact is sometimes useful: the kernel of a ring homomorphism is an ideal in the ring that is the domain of the homomorphism. (https://math.stackexchange.com/questions/1750840/kernel-of-ring-homomorphism-is-an-ideal) (Actually, the converse is true as well; Google for "quotient ring").

    Therefore, one approach to looking for ideals of a ring $R$ is to try constructing various homomorphisms from $R$ to other rings.

    I also suggest getting access to Introduction to commutative algebra by Atiyah and McDonald.

     – avs Apr 25 '19 at 23:59

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Throughout, I assume ideals was meant to mean two-sided ideals.

Rings (and their ideal structure) can be quite diverse, so it is not feasible to expect a comprehensive algorithm on determining the ideals of any ring.

Firstly, let's agree that it is relatively easy to understand a certain class of ideals: the principal ideals. At least there we know that for each element $x\in R$, there is an ideal $(x)$. In the case of commutative rings it is especially easy. In the case of principal ideal rings, that completely solves the problem of describing the ideals: the ideals correspond to elements of $R$ and not elements of $\mathcal P(R)$.

But after that, what do you do? Well, in principle you could just take arbitrary collections of elements and "check" the ideal they generate, but when you start using infinite subsets this seems problematic. At least, a concise picture of what they look like is quickly obscured by the combinatorial possibilities.

Another possibility is this: is the ring you are interested in a quotient of a ring whose ideals you understand already? If so, then you can make use of correspondence of ideals between the ring and its quotient. This is especially useful for quotients of PIDs like $\mathbb Z$ and $F[x]$ and $F[[x]]$.

I don't think I would advise this exhaustive search for ideals in anything other than a finite ring, and even then only one that a computer can handle.

One final bit of advice that might be useful: consider searching for central idempotents. The difficulty of doing this will depend on the ring you have chosen. But the idea is that each central idempotent decomposes the ring you have chosen into the product of two rings, and the ideal structure of the product of two rings has a simple description in terms of the ideal structures of the components.

If your ring has a complete set of orthogonal idempotents (what we call a semiperfect ring, you could quickly reduce the study of the ideals of this ring to the ideals of finitely many pieces of the ring, which are potentially easier to handle.

Suggested examples

  1. Find all the ideals of $\mathbb Z/30\mathbb Z$ both by the idempotent method and by the method of correspondence of ideals in the quotient ring.

  2. Find all the ideals of $M_2(F)$ by first considering all of its ideals generated by a single element. (In fact, you can find in many places on the site the complete characterization of ideals of $M_n(R)$ in terms of the ideals of $R$.)

  3. Find the ideals of some small rings exhaustively, such as $F_2[x,y]/(x^2,xy)$. Here are a bunch of finite rings you might try out, if they are not too big.

rschwieb
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