1.Simple factor algebras. 2.Radicals defined by simple algebras. 3.Radicals of finite dimensional algebras. 4.Stalks with zero radical. 5.Proposition. 6.Brown-McCoy radical. 7.Structure of module finite algebras. 8.Prime and semiprime. 9.The prime radical. 10.Weakly local algebras. 11.Uniserial algebras. 12.The weakly local radical.
13.Exercises.
The classical Wedderburn Structure Theorem for a finite dimensional associative algebra A over a field states that the factor algebra A/N by the nil radical N is a finite direct product of simple algebras. These simple algebras are matrix rings over division rings.
There were many attempts to extend this type of theorem to more general situa-tions. For associative rings A the Jacobson radical JacA can be defined as the sum of all quasi-regular (left) ideals andA/JacAis a subdirect product of primitive rings.
Again, a certain ideal is determined by properties of its elements and the factor ring by this ideal allows certain statements about its structure.
In general, looking for a radical X in any algebra A there are two points to be watched. On the one hand one has to care for the structure ofA/X, and on the other hand one would like to have an internal characterization of the ideal X.
Let us start with the first problem.
Definitions. An algebra without any proper non-zero ideals is calledquasi-simple.
A quasi-simple algebraA with A2 6= 0 is calledsimple.
An ideal K ⊂A is said to be modular if there existse∈A with e6∈K, a−ae∈K, a−ea∈K for every a∈A.
Obviously, the image of such an element e under A → A/K is the unit element in A/K. In a unital algebra every proper ideal is modular (with e= 1).
From the above definitions we obtain by 1.8:
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21.1 Simple factor algebras.
Consider an ideal K in an algebra A.
(1) The factor algebra A/K is quasi-simple if and only if K is a maximal M (A)-submodule (maximal ideal) of A.
(2) A/K is a simple algebra if and only if K is a maximal ideal with A2 6⊂K. (3) A/K is a simple ring with unit if and only if K is a maximal modular ideal.
The representation of algebras as subdirect products of factor algebras described in 1.10 yields our next results. As for associative rings, thedescending chain condition (dcc)on ideals is of importance:
21.2 Radicals defined by simple algebras.
Let A be an R-algebra.
(1) The radical of A as an M(A)-module is
Rad(A) = \ {K ⊂A|K a maximal ideal }, and Rad(A/Rad(A)) = 0.
Rad(A) = 0 if and only if A is a subdirect product of quasi-simple algebras.
IfAhas dcc on ideals, thenA/Rad(A)is a finite product of quasi-simple algebras.
(2) The Albert radical of A is defined as
Alb(A) =\ {K ⊂A|K a maximal ideal with A2 6⊂K}, and Alb(A/Alb(A)) = 0.
Alb(A) = 0 if and only if A is a subdirect product of simple algebras.
If A has dcc on ideals K with A2 6⊂ K, then A/Alb(A) is a finite product of simple algebras.
(3) The Brown-McCoy radical of A is defined as
BMc(A) = \ {K ⊂A|K a maximal modular ideal },
and BMc(A/BMc(A)) = 0. BMc(A) = 0 if and only if A is a subdirect product of simple algebras with units.
If A has dcc on modular ideals, then A/BMc(A) is a finite product of simple algebras with units.
(4) Jac(M(A))A⊂Rad(A)⊂Alb(A)⊂BMc(A).
If A2 =A, then Rad(A) =Alb(A).
If A has a unit, then Rad(A) =Alb(A) =BMc(A).
(5) Assume A is a finitely generated R-module, or R is a perfect ring.
Then Jac(R)A⊂Rad(A).
Proof. As mentioned above, most of the statements are immediately derived from 1.10. For the second part of (3) it is to check that, for any modular idealsU, V with U +V =A, U ∩V is also a modular ideal:
Assume e ∈A with e6∈ U, a−ae ∈ U, a−ea ∈U for a∈ A. It is easily verified that we may assume e ∈ V. Choosing similarly an f 6∈ V and f ∈ U the element e+f shows thatU ∩V is a modular ideal.
(4) These relations follow from the definitions and 6.16.
(5) This is a special case of 6.17. 2
For finite dimensional algebras we obtain as a corollary:
21.3 Radicals of finite dimensional algebras.
Let A be a finite dimensional algebra over a field K. Then:
(1) Rad(A) = 0 if and only if A is a direct product of quasi-simple algebras.
(2) Alb(A) = 0 if and only if A is a direct product of simple algebras.
(3) BMc(A) = 0 if and only if A is a direct product of simple algebras with unit.
Next we show that the above radicals of A are zero if all Pierce stalks have zero radical. Recall that we denote by X the set of all maximal ideals in the Boolean ring of idempotents ofR (see 18.1).
21.4 Stalks with zero radical. Let A be an R-algebra.
(1) If RadM(Ax)(Ax) = 0 for every x∈ X, then Rad(A) = 0.
(2) If Alb(Ax) = 0 for every x∈ X, then Alb(A) = 0.
(3) If BMc(Ax) = 0 for every x∈ X, then BMc(A) = 0.
Proof. (1)Ais a subdirect product of theAx by 18.1. Assume allRadM(Ax)(Ax) = 0.
Then the Ax are subdirect products of algebras without non-trivial ideals and the same is true forA, i.e., Rad(A) = 0.
The same argument yields (2) and (3). 2
21.5 Proposition. LetA be anR-algebra. The Brown-McCoy radical and the Albert radical of the R-algebra A are equal to the corresponding radical of the ZZ-algebra (ring) A.
Proof. We have to show that maximalZZ-idealsI ⊂AwithA2 6⊂I are alsoR-ideals.
Assume I is a maximal ZZ-ideal inA. ThenRI is anR- and aZZ-ideal containing I and hence RI =I orRI =A. However, the last equality would implyA2 ⊂RIA⊂ IRA⊂IA⊂I, contradicting the choice of I. 2 By construction the radicals defined above yield satisfying structure theorems for algebras with zero radical. However, so far we have no internal characterizations for these radicals. For the Brown-McCoy radical we can get such a characterazition similar to the Jacobson radical of associative algebras (see 6.16).
For an element a in any algebra A we define S(A) as the ideal of A generated by the elements ax−x+ya−y for all x, y ∈A, i.e.
S(a) := <{ax−x+ya−y | x, y ∈A}> ⊂A.
An elementa ∈A is calledS-quasi-regular if a ∈S(a).
An idealI ⊂A is said to be S-quasi-regular if every element of I is S-quasi-regular.
21.6 Brown-McCoy radical. Internal characterization.
(1) For any algebra A the Brown-McCoy radical BMc(A) is equal to the largest S-quasi-regular ideal.
(2) BMc(A) contains no central idempotents.
Proof. (1) Assumea∈Ais not S-quasi-regular, i.e. a6∈S(a). Then the set of ideals I ⊂Awith S(a)⊂I and a6∈I is non-empty and inductive (by inclusion). By Zorn’s Lemma it contains a maximal element, say J. We show that this is a maximal ideal inA.
Consider b ∈ A\J and the ideal < b > generated by b. Then, by maximality of J, a∈< b >+J and hence
a+ax−x∈< b >+J, for all x∈A.
This implies< b >+J =A. SinceS(a)⊂J,J is modular and hence J is a maximal modular ideal not containing a, i.e. a6∈BMc(A).
Now let Q be an S-quasi-regular ideal and K a maximal modular ideal in A.
Assume Q 6⊂ K. Then K +Q = A. By assumption, there exists e ∈ A\K with b−be∈K andb−eb∈K for everyb ∈A. Let us writee=k+q withk ∈K,q ∈Q.
Then
b−be=b−bq−bk∈K, and hence b−bq ∈K.
Similarly we obtain b−qb∈K. Since q is S-quasi-regular we conclude q ∈S(q)⊂K and e=k+q ∈K,
contradicting the choice of e. Hence Q⊂BMc(A).
(2) Assumee ∈BMc(A) is a central idempotent. Thene=ex−xfor some x∈A
implyinge =e2 =ex−ex= 0. 2
If the R-algebra A is finitely generated as R-module, then M(A) ⊂ Ak for some k∈IN (see 2.12). Hence the structure of M(A) is related to the structure of A:
21.7 Structure of module finite algebras.
Assume the R-algebra A is finitely generated as R-module.
(1) The following statements are equivalent:
(a) A is a finite direct product of quasi-simple algebras;
(b) M(A) is left semisimple.
(2) If R is artinian, then the following are equivalent:
(a) A is a finite direct product of quasi-simple algebras;
(b) Rad(A) = 0;
(c) M(A) is left semisimple;
(d) Jac(M(A)) = 0.
(3) If R is artinian, then:
(i) A/Rad(A) is a finite direct product of quasi-simple algebras.
(ii) M(A)/Jac(M(A)) is a left semisimple algebra.
(iii) Rad(A)Jac(M(A))·A.
Proof. (1) (a)⇒(b) Since M(A)⊂Ak and A is a semisimpleM(A)-module, M(A) is left semisimple.
(b)⇒(a) If M(A) is left semisimple, every left M(A)-module is semisimple.
(2) SinceR is artinian, Aand M(A) have dcc on (left) ideals. Hence (a)⇔(b) by 21.2 and (c)⇔(d) by the structure theory of associative left artinian rings.
(a)⇔(c) is shown in (1).
(3) (i) and (ii) are obvious by (2).
We always haveJac(M(A))·A⊂Rad(A). By (i),A/(Jac(M(A))·A) is a
semisim-ple M(A)-module and henceJac(M(A))·A⊃Rad(A). 2
Definitions. Let A be an R-algebra. A proper ideal P ⊂ A is called a prime ideal if, for any two ideals U, V ⊂A,U V ⊂P impliesU ⊂P orV ⊂P.
P is a semiprime ideal if U2 ⊂P impliesU ⊂P, for every ideal U ⊂A.
A is called aprime algebra if 0 is a prime ideal. It is called a semiprime algebra if the ideal 0 is semiprime, i.e., ifA has no non-zero ideals whose square is zero.
21.8 Prime and semiprime. Let A be an R-algebra.
(1) Every maximal ideal K ⊂A with A2 6⊂K is a prime ideal.
(2) Ais a semiprime algebra if and only if, for any idealsU, V ⊂A,U V = 0 implies U∩V = 0.
(3) If A is (semi-) prime, then the centroid C(A) is also a (semi-) prime algebra.
Proof. (1) Consider ideals U, V ⊂ A with U V ⊂ K. Assume U 6⊂ K and V 6⊂ K.
Then K+U =K+V =A and
A2 = (K +U)(K+V)⊂K, a contradiction.
(2) Assume A is semiprime and U V = 0. Then (U ∩V)2 ⊂ U V = 0 and hence U ∩V = 0. If the second property is given and U2 = 0, then U =U ∩U = 0, i.e., A is semiprime.
(3) Let A be prime. Consider two ideals I, J ⊂ C(A) with IJ = 0. Then (AI)(AJ) = A2IJ = 0 and hence AI = 0 or AJ = 0. This implies I = 0 or J = 0 showing that C(A) is prime.
Putting I =J we obtain our assertion for semiprime algebras. 2 Obviously, P is a (semi) prime ideal if and only if A/P is a (semi) prime algebra.
It is easy to verify thatAis (semi) prime asR-algebra if and only if it is (semi) prime as a ZZ-algebra (ring).
21.9 The prime radical.
For an R-algebra A the prime radical is defined as
P ri(A) :=\ {P ⊂A | P is a prime ideal }.
It has the following properties:
(1) P ri(A) is a semiprime ideal and P ri(A/P ri(A)) = 0.
(2) P ri(A) = 0 if and only if A is a subdirect product of prime algebras.
(3) P ri(A)⊂Alb(A).
Proof. (1) and (2) are immediate consequences of 1.10.
(3) This follows from the fact that any simple algebra is prime. 2 A word of caution is in order. An associative algebras A with unit is semiprime if and only if P ri(A) = 0 (e.g., [40, 3.13]). In general, P ri(A) = 0 still implies that A is semiprime. However, the converse conclusion need no longer be true.
Definitions. LetA be an R-algebra. An idealL⊂A is called weakly localif it is contained in a unique maximal ideal K ⊂A with A2 6⊂K.
A is called a weakly local algebra if 0 is a weakly local ideal.
Of course, every simple algebra is weakly local. The ring R is weakly local if and only if it is local in the classical sense. In general we have the following characteriza-tions:
21.10 Weakly local algebras.
For an R-algebra A the following are equivalent:
(a) A is a weakly local algebra;
(b) A is a local M(A)-module and Rad(A) =Alb(A);
(c) Rad(A) is a maximal ideal and A2 6⊂Rad(A);
(d) A/Rad(A) is a simple algebra;
(e) A is finitely generated as M(A)-module, and
every factor algebra B of A is indecomposable and B2 6= 0.
Proof. (a) ⇒ (b) Assume K is the only maximal ideal in A and A2 6⊂ K. Then K =Alb(A) =Rad(A).
(b)⇒(c)Alb(A) is a maximal ideal and hence A2 6⊂Rad(A).
(c)⇔(d) This is clear from the definitions.
(c)⇒(e) As a localM(A)-module,Ais finitely generated and every factor algebra is indecomposable (see 8.2). The kernel of every surjective algebra morphismA→B is contained inRad(A) and A2 6⊂Rad(A) implies B2 6= 0.
(e)⇒(a) Consider two maximal idealsK1, K2 ⊂A. Since (A/Ki)2 6= 0,A2 6⊂Ki. AssumeK1 6⊂K2. Then A/(K1∩K2)'A/K1⊕A/K2 is a non-trivial decomposition
of a factor algebra, a contradiction. 2
Recall that a moduleM is calleduniformif every submodule is essential in it, and M is uniserial if its submodules are linearly ordered by inclusion.
21.11 Uniserial algebras.
For an R-algebra A the following statements are equivalent:
(a) The ideals in A are linearly ordered by inclusion;
(b) every factor algebra B of A is a uniform M(B)-module.
Under these conditions, an algebra A with unit is weakly local.
Proof. (a)⇒(b) Factor modules of uniserial modules are uniserial, hence uniform.
(b) ⇒(a) Every factor algebra B of A has simple or zero socle as an M(B)- and M(A)-module and hence is uniserial by 8.15.
If A has a unit, Rad(A) is a unique maximal ideal. 2 Based on the notions considered above we introduce another radical:
21.12 The weakly local radical.
For an R-algebra A the weakly local radical is defined as
Loc(A) := \ {L⊂A | L is a weakly local ideal }.
It has the following properties:
(1) Loc(A/Loc(A)) = 0 and Loc(A)⊂Alb(A).
(2) Loc(A) = 0 if and only if A is a subdirect product of weakly local algebras.
(3) If A2 = A and A is finitely generated as M(A)-module and has dcc on ideals, then A/Loc(A) is a finite product of weakly local algebras.
Proof. (1), (2) follow immediately from the definitions and 1.10.
(3) We may assume Loc(A) = 0. Choose a minimal set of weakly local ideals Li withL1∩ · · · ∩Lk= 0 and denote byKi the unique maximal ideal of AwithLi ⊂Ki. Suppose Ki = Kj for some i 6= j. Then Li ∩Lj is a weakly local ideal: Assume Li∩Lj ⊂K for some maximal ideal K 6=Ki. Then LiLj ⊂K and hence Li ⊂ K or Lj ⊂K, sinceK is a prime ideal. HenceLi orLj is contained in two maximal ideals, a contradiction.
Therefore, by our minimality assumption, we have Ki 6=Kj for i6=j and Li+Lj is not contained in any maximal ideal of A. Hence Li +Lj = A and the assertion
follows from the Chinese Remainder Theorem 1.10. 2
21.13 Exercises.
(1) Let A be an associative algebra. Show for matrix rings (n∈IN):
BM c(A(n,n)) = (BM c(A))(n,n).
(2) LetA be an associative semiperfect ring with unit. Assume {e1, . . . , en} to be a complete set of primitive central idempotents modulo Jac(A). Prove that ([178])
Loc(A) = X
i6=j
AeiAejA.
(3) Let (A,+,·) be a simple associative ring. Prove ([247]):
(i) Assume S ⊂Ais a subring such that S2 6= 0 and SAS=S. ThenS has a unique maximal ideal
{t∈S|StS = 0}.
(ii) Leta ∈A such thata2 6= 0. Then the ring aAahas a unique maximal ideal {s∈aAa|asa= 0}.
(iii) Letc∈A such that AcA6= 0. Define a new product on (A,+) by
∗:A×A→A, (a, b)7→acb.
Then the ring (A,+,∗) has a unique maximal ideal {a∈A|cac= 0}.
(4) Let A be an associative ring with unit which has a unique maximal ideal M.
Prove ([240, Theorem 3]):
A is either a local ring or every maximal right ideal I 6= M is an idempotent non-two-sided ideal.
Furthermore, (IJ)2 =IJ for any maximal right ideals I, J different from M. References: Albert [43], Behrens [68, 69], Brown-McCoy [91], Gray [17], Jenner [167], Kerner-Thode [178], Pritchard [223], Satyanarayana-Deshpande [240], Schafer [37], Stewart-Watters [247], Zwier [289].