Download PARTING THOUGHTS ON PI AND GOLDIE RINGS 1. PI rings In this

PARTING THOUGHTS ON PI AND GOLDIE RINGS
1. PI rings
In this section we’ll define PI rings (a class of rings generalizing commutative
rings) and state a few important results regarding them.
Definition 1.1. [MR, 13.1.2] Let f ∈ Zhx1 , . . . , xn i for some n. The degree of f
is the maximum length of a monomial in f . The polynomial f is monic if at least
one of the monomials of maximum length has coefficient 1.
Definition 1.2. [MR, 13.1.6] Let R be a ring. Then R is a polynomial identity (PI)
ring if there exists n and f ∈ Zhx1 , . . . , xn i, with f monic, such that f (a1 , . . . , an ) =
0 for all ai ∈ R. The polynomial f is a polynomial identity for R.
Remark 1.3. The requirement that a polynomial identity f be monic precludes
some badly behaved rings, such as R = khx, yi, where k is a field of characteristic
p > 0. If f (z) = pz, then f (r) = 0 for all r ∈ R, but R does not behave like a
commutative ring.
It can be shown that if R is a PI ring, then R satisfies a multilinear polynomial
identity [MR, 13.1.9]. That is, a polynomial where all the monomials are length n
permutations of x1 , . . . , xn .
Of course any commutative ring satisfies xy − yx. This can be generalized to
matrix rings in the following manner.
Definition 1.4. [MR, 13.1.3] Let n > 0, and let x1 , . . . , xn be variables. Then the
nth standard (polynomial) identity is given by
X
sn =
sign(σ)xσ(1) xσ(2) . . . xσ(n) ,
σ∈Sn
where Sn is the symmetric permutation group on n elements and σ is a permutation.
Theorem 1.5 (Amitsur-Levitzki). [MR, 13.3.3] If A is a commutative ring, then
Mn (A) satisfies s2n . Therefore Mn (A) is a PI ring.
One of the amazing facts about PI rings is that many questions can be reduced
to questions about matrix rings and the standard identities sd , due to the following
embedding theorem.
Theorem 1.6. [MR, 13.4.2] Let R be a semiprime PI ring, and let d be the minimum of the degrees of the polynomial identities of R. (It can be shown that d is
even [Ro, 1.6.27].)
Then there exists a commutative ring A (which is a product of fields), such that
R embeds as a subring of Mn (A) for n = (d/2)!. If R is prime, then A can be
chosen to be a field.
If one does not require that 1 ∈ R maps to 1 ∈ Mn (A) in the embedding, then
one can assume that n = d/2. Thus sd is a PI for R by (1.5).
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PARTING THOUGHTS ON PI AND GOLDIE RINGS
It was mentioned in class that prime PI rings are Goldie. Actually, a stronger
result is known.
Theorem 1.7 (Posner). [MR, 13.6.5] Let R be a prime PI ring with center Z.
(We have seen that Z is a domain.) Let Q(Z) be the quotient field of Z. Then R
is a Goldie ring with maximal quotient ring Q(R) such that Q(R) = RQ(Z), the
center of Q(R) is Q(Z), and Q(R) is finite dimensional over Q(Z) (i.e., Q(R) is
a central simple algebra).
So for example, Posner’s theorem tells us that the quotient ring of Mn (Z) is
Mn (Z)Q, which is easily seen to be Mn (Q).
One of you asked which semiprime commutative rings are Goldie. Of course an
infinite direct product R of fields is semiprime, but contains an infinite direct sum
of ideals, so R is not Goldie. Another problem with such an R is the following.
Proposition 1.8. [Lec, 11.22] Let R be a semiprime right Goldie ring, let Q be
the semisimple maximal right quotient ring of R, and let Pi be the (finitely many,
minimal) prime ideals of Q. Then the minimal prime ideals of R are exactly the
Pi ∩ R. In particular, R has finitely many minimal prime ideals.
Proposition 1.9. [MR, 3.2.5] Let R be a semiprime ring with finitely many minimal prime ideals Pi , i = 1, . . . , n. Then R is right Goldie if and only if each R/Pi
is right Goldie, i = 1, . . . , n.
The previous two propositions then allow us to characterize all semiprime commutative (even PI) Goldie rings.
Corollary 1.10. Let R be a semiprime PI ring. Then R is Goldie if and only if
R has finitely many minimal prime ideals.
Proof. For ⇒, use (1.8). For ⇐, use (1.7) and (1.9).
2. Modules over semiprime Goldie rings
In class I showed how simple right modules of right noetherian, right primitive
rings R could be thought of as Q(R)-modules where Q(R) is the simple artinian
quotient ring of R. This was just one example of a larger theorem. First, I’ll remind
you of a few definitions.
Definition 2.1. Let R be a semiprime right Goldie ring and let M be a right
R-module. Then τ (M ) = {m ∈ M |∃x ∈ R, x regular, such that mx = 0} is the
torsion submodule of M . (If R is not semiprime right Goldie, this subset may not
be a submodule [GW, Exercise 6C].) If τ (M ) = M , then M is torsion. If τ (M ) = 0,
then M is torsionfree. The module M is divisible if M = M x for all regular x ∈ R.
A word of warning: torsion can take on different meanings in different contexts.
For example, for graded modules over graded rings, a torsion module is one in which
for each m ∈ M , there exists n such that mR≥n = 0, where R≥n = ⊕∞
i=n Ri .
Proposition 2.2. [GW, Prop. 6.13] Let R be a semiprime right Goldie ring, and
let R be a right order in the semisimple ring Q.
(1) If M ∈ Mod −Q, then M is torsionfree and injective as a right R-module.
If N is a right R-submodule of M , then N is a right Q-module if and only
if N is a divisible R-module.
PARTING THOUGHTS ON PI AND GOLDIE RINGS
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(2) Every torsionfree divisible right R-module has a unique right Q-module
structure compatible with its right R-module structure.
(3) A right Q-module is uniform as an R-module if and only if it is isomorphic
to a minimal right ideal of Q.
References
[GW] K. R. Goodearl and R. B. Warfield, Jr., An introduction to noncommutative Noetherian
rings, Cambridge University Press, Cambridge, 1989. MR 91c:16001
[Lec] T. Y. Lam, Lectures on modules and rings, Springer-Verlag, New York, 1999. MR 99i:16001
[MR] J. C. McConnell and J. C. Robson, Noncommutative Noetherian rings, revised ed., American Mathematical Society, Providence, RI, 2001, With the cooperation of L. W. Small. MR
2001i:16039
[Ro] L. H. Rowen, Polynomial identities in ring theory, Academic Press Inc. [Harcourt Brace
Jovanovich Publishers], New York, 1980. MR 82a:16021
E-mail address: [email protected]
URL: http://www.mit.edu/~dskeeler/706/