Primary Ideals, Primary Module

Primary Module

You thought your head was spinning before - hang on to your hat. A module can be primary, just as an ideal is primary, and many of the theorems in this topic can be generalized to modules.

Given an R module M and an element y in R, call y a zero divisor if the module endomorphism y*M is not injective. In other words, yx = 0 for some nonzero x in M.

Call y nilpotent if yⁿ kills M. (Note that yⁿ could be 0; that's ok.) Thus the endomorphism y*M, invoked n times, produces 0. Clearly y is a zero divisor.

A proper submodule H in M is primary if every 0 divisor of M/H is nilpotent in M/H. Let's provide an equivalent definition in terms of x and y. Let y be a zero divisor, so that xy lies in H for some x outside of H. Now some yⁿ drives M into H.

Let M = R, so that H becomes an ideal, and review the above definition. If y is in H it becomes 0 in R/H. If y and x are outside of H, with xy in H,then some power of y drives R into H. Since R contains 1, some power of y lies in H. Therefore H is a primary ideal. The definitions are consistent.

Let H be a primary submodule of M, and let Q be the set of elements of R that drive M into H. Note that Q is an ideal, in fact it is a proper ideal, since Q does not contain 1.

Let xy lie in Q, so that xyM lies in H. Assume y does not lie in Q. thus yz is not in H for some z in M. Since x drives yz into H, x becomes a zero divisor, and is nilpotent. Some power of x drives M into H, and xⁿ lies in Q. Therefore Q is a primary ideal.

If M = R, only H can drive 1 into H, whence Q = H.

continuing the above, let H be a P primary module if P = rad(Q). Thus y is in P iff some power of y drives M into H.

A proper submodule J has a primary decomposition if it is the finite intersection of primary submodules. This decomposition is reduced if none of the primary modules contains the intersection of the others, and all the prime radicals are distinct.

Let H₁ H₂ H₃ etc be a finite set of P primary modules, and let J be their intersection. If yⁿ takes M into J it takes M into each H_i. Conversely, if some yⁿ takes M into each H_i, select the largest n, and yⁿ takes M into J. The radical of J is P.

Let y be a zero divisor in R/J. If yx lies in J and x is not in J, x is not in at least one of the primary modules H_i. A power of y drives M into H_i, y lies in P, and a (possibly higher) power of y drives M into J. Therefore J is a P primary module.

Replace a set of P primary modules with their intersection, hence each prime radical is represented only once. Remove any redundant modules, that contain the intersection of the others. The result is a reduce primary decomposition. A primary decomposition implies a reduced primary decomposition.

If H is a submodule and x is in M, the conductor ideal [H:x] is the ideal in R that drives x into H. Note that x is in H iff [H:x] = R. (Remember, modules are assumed unitary, so 1*M = M.)

Assume H is primary, yz is in [H:x], and z is not. With yzx in H, y is a zero divisor, y is nilpotent, yⁿ drives M into H, yⁿx lies in H, and yⁿ is in [H:x]. This makes [H:x] a primary ideal.

The radical of [H:x] includes y iff yⁿx lies in H. this is contained in rad(H), which drives all of M into H. Conversely, if yx lies in H then yⁿ drives M into H, H being primary. For any x outside of H, rad([H:x]) = rad(H) = P.

If J is the intersection of several modules H_i, take a moment to show [J:x] is the intersection of [H_i:x]. Beyond this, rad([J:x]) is the intersection of rad([H_i:x]).

Review the proof of first uniqueness; it applies here. If J has a primary decomposition, x ∈ M can be in none, some, or all of the primary modules H_i. When x is in all but one of them, missing H_i, rad([J:x]) produces P_i, the radical of H_i. If x misses multiple primary modules, rad([J:x]) becomes the intersection of several prime ideals. If this is prime it has to equal one of the prime ideals. Therefore the prime radicals over the various conductors of J, denoted σ(J), are the prime radicals over the primary modules in the reduced decomposition of J. This depends only on the structure of M, hence the primes in the reduced decomposition are fixed. First uniqueness extends to modules.

Second uniqueness can also be generalized to modules, and this can be used to characterize laskerian modules, where every proper submodule has a primary decomposition. I'm going to skip the details. It looks like a lot of work, and I think you get the idea. Besides, laskerian modules don't come up very often. I may expand this section in the future, if warranted.