SIMPLE AND SEMISIMPLE NEAR-RINGS … · A near-ring homomorphism is a mapping t of a near-ring N...

SIMPLE AND SEMISIMPLE NEAR-RINGS

D. W. BLACKETT

Algebraic systems with addition and multiplication which satisfy

all of the skew-field axioms except possibly one of the distributive

laws have been studied occasionally. Dickson [l] gave examples

which showed that such systems—called Fastkörper by Zassenhaus—

can actually be nondistributive. Zassenhaus [2] proved that Dick-

son's examples include all but seven of the finite Fastkörper which

are not skew-fields. All continuous Fastkörper of finite degree over

the reals were found by Kalscheuer [3]. Reidemeister [4] connected

Fastkörper with the geometry of webs. All these results are, in a

certain sense, extensions of well known facts about fields. This paper

makes an analogous extension of part of the theory of semisimple

rings to semisimple "Fastringe" or "near-rings."

Definition. A set N of elements which can be added and multi-

plied is said to form a near-ring if

1. the elements form a group under addition,

2. the multiplication of elements is associative,

3. Mi(m2+w3) =Mim2+Mi«3 for any selection of elements «1, n2, and

n3 from N.

It follows from this definition that a near-ring satisfies all of the

usual ring axioms with the possible exceptions of the right distribu-

tive law and the commutative law of addition.

The most natural example of near-rings is given by the mappings of

a group (written additively) into itself. If the mappings are added

by adding images and multiplied by iteration they form a near-ring. '

A near-ring homomorphism is a mapping t of a near-ring N into a

near-ring N' such that

(»1 + n2)r = «it + n2r and (wiw2)r = (wir)(«2r).

The two-sided ideals of a near-ring N are defined to be the kernels

of homomorphisms of N. The kernel A of a near-ring homomorphism

t of N is an additive normal subgroup of N. Since near-rings satisfy

the left distributive law,

Presented to the Society, September 3, 1952; received by the editors March 6,

1953.1 For a more complete description of this example see H. Zassenhaus, Lehrbuch

der Gruppentheorie, Berlin, 1937, pp. 71-74, or N. Jacobson, The theory of rings, New

York, American Mathematical Society, 1943, pp. 1-2.

772

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SIMPLE AND SEMISIMPLE NEAR-RINGS 773

»0 = 0

when « is any element of N and 0 is the additive identity of N.

Therefore NKEK. Also

[(»i + A)»2 — «i«2]t = [(«i + k)n2]r — («i»2)r

= [(»i + A)t](«2t) — (»i»2)t

= (»it + At)(»2t) — (»i«2)r

= (»it)(m2t) - (»ir)(«2r)

= 0

for «i and »2 in N and A in K. Thus (»i+A)«2—»i«2 is in K for every

selection «i and «2 from N and A from K.

Suppose now that H is any additive normal subgroup of N such

that NHEH and (»i+A)«2 —»i«2 is in H for «i and w2 in N and A in

H. If a is the natural group homomorphism of N onto N—H, a simple

calculation shows that the definition

(»i + H)in2 + H) = »i«2 + H

makes N—H a near-ring and allows <r to be considered a near-ring

homomorphism.

Thus the two-sided ideals of N are just the additive normal sub-

groups T of N such that

1. NTET,2. («i+i)«2—»i«2 is in T if »i and »2 are in N and t is in T.

An additive group G with the elements of N as right operators is a

right AT-group if

(g»l)»2 = S(«l»2)

for every selection of »i and «2 from N and g from G. If "operating by

»" is defined as right multiplication by », a near-ring N is itself a

right AT-group. The additive subgroups of N which are also right

A7-groups are called the right modules of N. These modules are just

the subgroups M such that MNEM.

A group homomorphism t of a right A'-group G into a right N-

group is a right AMiomomorphism if

ign)r = igr)n

for every g and ».

The right ideals of N are defined to be the kernels of homo-

morphisms of N regarded as a right A-group. A check shows that

the right ideals are just the additive normal subgroups R of N with

(r + »i)»2 — »i»2 E R


774 D. W. BLACKETT [October

for every m, n2, and r.

For a ring the set of right modules is identical with the set of right

ideals. This is not true in general for near-rings. The near-ring of all

mappings of a finite group (with order greater than two) into itself is

an example where the whole near-ring N is the only right ideal which

is also a right module. Consideration will shortly be restricted to those

near-rings such that Ok = 0 for all n. For these near-rings the set of

right modules contains the set of right ideals.

Left A-groups and left A-homomorphisms may be defined analo-

gously. If "operating by n" is left multiplication by », JV is a left

A-group. The left ideals of A are defined to be the kernels of homo-

morphisms of A considered as a left A-group. These ideals are just

the additive normal subgroups L of A with NLEL. In terms of

these definitions, a two-sided ideal of A is just a right ideal which is

also a left ideal.

A right A-group V is called a right representation space of N or

simply a right A-space if

v(ni + w2) = vni + vn2

for every «i, w2, and v. A right A-space is proper if FA^O. An ir-

reducible right A-space is defined as a proper A-space with no sub-

spaces other than the whole space and the space containing only 0.

The right modules of A are examples of right A-spaces. Since a right

A-homomorphism of a right A-space V automatically preserves addi-

tion of operators, representation space homomorphisms are just the

right A-homomorphisms which carry one representation space into

another.

The sets of left A-groups, right A-groups, right modules, left

ideals, right ideals, two-sided ideals, and right representation spaces

of a near-ring A are all closed under finite intersection. If the sum of

two left, right, or two-sided ideals is defined as the set of all sums with

a summand from each of the ideals, the sets of left, right, and two-

sided ideals are closed under finite sums. This is not necessarily true

of the right modules. An A-group, an ideal, or a representation space

is a direct sum of other A-groups, ideals, or representation spaces if

it is a direct sum of them when considered as an additive group.

A simple calculation shows that the elements of a near-ring A

which annihilate a right A-space F form a two-sided ideal Ay of A.

Similarly the set of elements of A annihilating one particular element

» of F is a right ideal Av.

From this point on all the near-rings considered will be assumed

to satisfy


'9531 SIMPLE AND SEMISIMPLE NEAR-RINGS 775

Axiom. 0» = 0 for all » in the near-ring.

Definition. A near-ring N is semisimple if it has no nonzero nil-

potent right modules and the right modules satisfy the descending

chain condition.

Theorem 1. Every nonzero module of a semisimple near-ring N

contains a nonzero idempotent. Every minimal nonzero right module M

is an irreducible N-space and contains an idempotent e such that

eN = eM = M.

Because the descending chain condition is satisfied by the right

modules of N, any nonzero module contains a minimal nonzero

module. Therefore, the first statement of the theorem is a conse-

quence of the second. If M is a minimal nonzero module, MMÔ

since N contains no nilpotent nonzero modules. This shows that M

is a proper A-space and hence an irreducible A-space. Since M2?*0,

there is an element w0 of M such that moMÔ. The mapping

t : m-^mom of M into itself is an A-homomorphism under which the

image of M is not zero. The kernel of the homomorphism is a right

module because of the assumption 0» = 0 for all ». The minimal prop-

erty of M implies that m0M=M and that Am/~\M is zero. There-

fore r is an A-isomorphism of the module M onto itself. Now there is

some element e in M such that moe = m0. Hence, m0ee = moe. It follows

that e2 = e because r is an isomorphism. Now eMEeNEM because

eEM. Since the module eM contains the nonzero element e of the

irreducible space M, MEeM. Therefore eM=eN= M.

It is to be noted that if e is any idempotent and x is an element of

eN, then ex = x. This follows because x = e» implies ex = een = en=x.

Theorem 2. A minimal nonzero right ideal R of a semisimple near-

ring N is an irreducible right representation space of N.

Because 0» = 0 for all « in N, R is a right module. If M is some

minimal nonzero module contained in R and m0 is some element of M

such that moMÔ, then the mapping r: n—*m0n is an A-homo-

morphism of A as a right A-group onto M. The kernel of t is the

right ideal Amo of annihilators of mo. Since R is a minimal right ideal,

either Rr\Am,=R or Ri\Am, = 0. Now RDM, but Amt$M. Hence

Rr\Am„ = 0. Therefore r restricted to R is an A-isomorphism of R

onto M. But Mr = M. Hence, R = M. By Theorem 1, M is an irreduc-

ible A-space.

Theorem 3. A nonzero right module M of a semisimple near-ring N

is a finite direct sum of minimal nonzero modules contained in it.



By the descending chain condition, M contains a minimal nonzero

module M0. Theorem 1 shows that an idempotent element e0 may

be chosen such that e0A=eoM0 = Mo. If m is any element of M,

m = eam + (— e0m + m).

Here eom is in Mo and — e0m-r-m is in AH. The latter follows because

2eo(— eom + m) = — eom + eom = — e0*» + eom = 0.

Therefore MEMo+Aei¡. Also Af</^.deo = 0 because eom0 = mo tor m0

in Jlf0. If M= Mo, the theorem is proved. If M^M0, then AOilfPL4«,0

5^0, that is, if properly contains a nonzero submodule which is the

intersection of M with a right ideal of A. Similarly either MC\Aet is

a minimal nonzero module or it properly contains a smaller nonzero

submodule of the form (MI^A^i^R where R is a right ideal of N.

Since Aeoi~\R isa right ideal of A, Mf\(Aeiir\R) is the intersection of

M with a right ideal of A. Because of the descending chain condition

on right modules this process must lead after a finite number of steps

to a minimal nonzero module Mi which is the intersection of M with

a right ideal of A. In particular this means that Mi is an additive

normal subgroup of M. After the selection of an idempotent d in

Mi, such that eiA=CiMi = lfi, M can be written as the direct sum

Mi®(Mf\Aei), because both Mi and (Mii\Aei) are normal sub-

groups of M. Repetition of the procedure with (Mi(~\A ei) and succeed-

ing groups shows that M=Mi®M2® • • • ®M, where the M i are

minimal nonzero modules contained in M. The chain condition as-

sures that this decomposition is of finite length.

If the additive group of A is abelian, the module Ma could serve

as a starting point of the decomposition, and the proof of the theorem

would have been simplified.

Corollary 1. A nonzero right ideal Rof a semisimple near-ring N is

the direct sum 'of minimal nonzero right ideals contained in it.

If the module M in the proof of the theorem happens to be a right

ideal, Mi, Mi, • • • , M, are also right ideals.

This remark, which proves the corollary, also gives a new proof of

Theorem 2.

Theorem 4. Every right ideal Rof a semisimple near-ring A contains

an idempotent e such that eR = R.

The procedure used in proving Theorem 3 allows the assumption

that R is represented as the direct sum R = exR ® • • ■ ®epR where c<


19531 SIMPLE AND SEMISIMPLE NEAR-RINGS 777

is an idempotent, e,7? = e,N is a minimal right ideal, and e,2? © • • •

(BepR is in A„,_,.

First it will be proved by induction that

iex + e2 + ■ • ■ + eP)R = e,P © e2R © • • • © epR = R.

Assume as an induction hypothesis that

(ei + • • • + e,)R = exR © • • ■ © e.R.

Since the summands e¡R of R are all right ¡deals,

iex + ■ ■ ■ + e, + e,+i)r -(«!+■•• + e,)r

is in e.+iR and

(ei +•••+«. + «.+!)»• - C+ir

is in di?© • • • ffie.P,. Therefore

(d + • • • + e, + e,+i)r — (ci + • • • + e,)r - e,+1r

is in

iexR + ■■■ +e,R)r\ e.+xR.

Because this intersection contains only zero,

(«i +•••+«» + e,+x)r = (ei + • • • + e.)r + e,+1r.

Repetition of this argument shows that

(«i + • • • + C + e,+x)r = exr + ■ • • + e.r + e,+1r

for any element r of R.

Now

iex + • • • + e, + e,+x)e.+xr = e,+1r

because et+lR is in Ati for i^s. Therefore («i+ • • • +e,+1)R contains

e,+1R.

On the other hand

(«!+•••+ e,+x)ir - e1+1r)

= (ci + • • • + e,+i)r — id + • • • + e,+x)e,+1r

= («i + • • • + e,)r + e,+lr - e,+lr

= (ei + • • • + e,)r.

Hence (ci+ • • • +e,+x)R contains (ci+ • • • +e,)R.

These facts, combined with the assumption that



(«!+•••+ e,)R = exR © • • • © e.R,

prove that

iex -\-+ e,+x)R = exR © • • • © e.+1R.

Since exR = exR, it follows by induction that

iex -\-+ ep)R = exR © • • • © epR = R.

It now follows that the right homomorphism of A defined by

r: «—>(ci+ • • • +ep)n carries R onto itself.

If

r = r, + r,+x + ■ • ■ + rp (r< E e(R.)

is an element of R such that r.^O, then

(«i + • • • + ep)r

-(«!+••• + «,)(r. + • • • + rp)

= Ci(r. + • • • + rp) + • ■ ■ + e.ir, + • ■ ■ + rp) + • ■ •

+ ePir. + •••+*•„)

= 0 + e,ir. + ■ ■ ■ + rp) + e.+xir. + • • ■ + rp) + ■ ■ ■

+ ePir, + • • • +rp)

= e,r, + e,ir,+x + ■ ■ ■ + rp) + e.+xir. + ■ ■ ■ + rP) + ■ • •

+ ePir. + ■ ■ ■ +rp)

= e,r, + e,+xir, + ■ ■ ■ + rp) + ■ ■ ■ + «p(r, + • • • + rp).

Since e.r^r.T^O and (ci+ • • • +ep)r—e.r, is in e.+iiî© • • • @epR,

(«,+ •••+ ep)r 9* 0.

This proves that r acts as an automorphism on R. Il e is the element

of R such that

iex + • • • + eP)e = ex + • • • + ep,

then et = e, and eR = R. This proves the theorem.

Corollary 1. Every semisimple near-ring has a left unit.

By Theorem 4 the near-ring has a generating idempotent. On the

other hand, any idempotent e is a left unit for eN.

Corollary 2. If R = Rx@ • ■ • @RPis a direct sum decomposition

of a right ideal R into smaller right ideals Rit generating idempotents e¡

can be selected from the respective R( so that c,d = 0/or ii*j.


i953l SIMPLE AND SEMISIMPLE NEAR-RINGS 779

By Theorem 4, R has an idempotent e such that eR = R. This

idempotent, which must act as a left unit on R, has a decomposition

e = d+ • • • +ep (eiERi)- As in the proof of Theorem 4,

rf = er¿ = (ei + • • • + ep)r¿ = cif¿ + • • • + epr{

tor any r< in R{. This shows that eyr¿ = 0 lor jî and that e%ri = r{.

Corollary 3. TAe rigft/ ¿¿ea/s o/ a semisimple near-ring satisfy the

ascending chain condition.

Suppose .R1C.R2C.RsC • • • is an ascending chain of distinct right

ideals. Let JR = U,J?¿. Select idempotents c,- such that etERi and

c,R,=i?,-. Now Ri+i=Ri®(Ri+ir\A,i). Therefore

R = Ri ® R{

where

Rf = (Ri+ilÂ.,) ® (Ri+trÂ.i+1) ® ■ • • .

Since the properly descending chain

RC D R2' D R¿ D ■ • •

must be finite, the original ascending chain must also be finite.

Theorem 5. If N is a semisimple near-ring and V is an irreducible

right N-space, V is N-isomorphic to a minimal nonzero right ideal of N.

In particular V is N-isomorphic to any minimal nonzero right ideal

Rfor which VRÔ.

Because irreducible spaces are proper, FAÔ. Therefore it follows

from Corollary 1 of Theorem 3 applied to A that FRpÔ for some

minimal nonzero right ideal R. If v is an element of F such that

vRt¿0, the mapping r: r—*vr is an A-homomorphism of R into F.

Since V is irreducible, r maps R onto F. The kernel RC\AV of t is 0

because R is a minimal nonzero right ideal. Hence V is A-isomorphic

to-R.2

Theorem 6. A semisimple near-ring has only a finite number of

non-N-isomorphic types of irreducible spaces. The number of types is

the same as the number of types of minimal nonzero right ideals which are

not N-isomorphic.

Theorems 2 and 5 show that the number of types of irreducible

spaces is the same as the number of types of minimal nonzero ideals.

It only remains to prove that this number is finite. The common

* A suggestion of the referee simplified this proof.



proofs of the Jordan-Holder Theorem for groups with linear opera-

tors carry over almost verbatim to proofs of the analogous theorem

for right ideals of a semisimple near-ring. This theorem allows the

normal series ADiOO iR a nonzero right ideal) to be refined to a

finite composition series. If R is a minimal right ideal, it must be a

factor of this composition series. Since all the composition series have

the same factors, the number of non-A-isomorphic types of minimal

right ideals is finite.

Definition. A nonzero near-ring N is simple if

1. it has no proper two-sided ideals,

2. its right modules satisfy the descending chain condition,

3. it has no nonzero right modules which are annihilated from the

right by all the elements of A.

Theorem 7. A simple near-ring is semisimple and has one and only

one type of irreducible space.

By the definition of simplicity, the right modules satisfy the de-

scending chain condition. If M is a nonzero nilpotent module, there

is some integer A such that M* = 0 but Mk~l^0. Therefore, the two-

sided ideal Am of annihilators of M contains Mk~19é0. This implies

that Am = N, which contradicts Condition 3 above. Thus A is a

semisimple near-ring. Since A is not the near-ring consisting of zero

alone, A must have a minimal nonzero module and hence at least one

type of irreducible A-space. If Fi and V2 are two irreducible A-spaces,

neither of them can be annihilated by a minimal right ideal R of A,

because the two-sided ideal of annihilators of either Fi or V2 must

be the zero ideal. By Theorem 5 both Fi and F2 are A-isomorphic to

R and hence to each other.

Theorem 8. A semisimple near-ring A with exactly one type of

irreducible N-space is a simple near-ring.

The right modules satisfy the descending chain condition because

A is semisimple. There can be no nonzero right module annihilated

by A since such a module would satisfy the equation M2 = 0. If F is a

proper two-sided ideal of A, there is at least one minimal right ideal

not contained in T, because a semisimple near-ring is the direct sum

of some of its minimal right ideals. If R is such a right ideal, R(~\T=0

because R is minimal. Also RTERf^T. Hence RT=0. Now R and

any minimal right ideal R' contained in T are A-isomorphic because

they are both irreducible A-spaces. Therefore R'R'ER'T=0, and

R' is nilpotent. This contradiction shows that A is simple.

For the rest of this paper the term "semisimple near-ring" will be


1953) SIMPLE AND SEMISIMPLE NEAR-RINGS 781

interpreted so as to exclude the trivial near-ring with only one ele-

ment.

If A is a semisimple near-ring, Theorem 6 permits the choice of

non-A-isomorphic irreducible A-spaces Vi, • • • , V. so that any other

irreducible A-space is A-isomorphic to one of the F,-. After such a

selection, a two-sided ideal A* corresponding to F< may be defined by

Ai = fi Av¡.Mi

Cleariy this definition depends only on the type of irreducible space

and not on the particular choice of the representative F<.

Lemma 1. A minimal right module is contained in A* if and only if it

is N-isomorphic to Vi.

A minimal right module M which is A-isomorphic to V¡, where

jy¿i, is annihilated by A° since A°EAr¡. Therefore, M cannot be con-

tained in A°t because M would then be nilpotent. If M is a minimal

module A-isomorphic to Vi, F,Af=0 for J9*i. This is the case, for

otherwise the mapping m—tVj-m would be an A-isomorphism of M

onto Vi for any v¡ such that v¡Mj^O. This shows that MEAv¡ for all

J9¿i. Therefore MEA°.

Theorem 9. A° is the sum of the modules N-isomorphic to Vi.

This follows immediately from Theorem 3 or its corollary, and

Lemma 1.

Lemma 2.

AlniZA0)

( Z a°)a°\ Mi /

From the definition of A\,

A\r\Z£- f\Avk-Mi *-l

If ni-i-4vt were not the zero ideal, it would contain a minimal non-

zero right ideal. This ideal would be A-isomorphic to one of the Vk

and accordingly would be nilpotent. Therefore

¿,n£¿, = 0.Mi

= 0,

i = 1, 2, ■ ■ ■ , s.

= 0,



Since E/*« -^y and -¿î are two-sided ideals,

(E/î)^c^n(E^).

Theorem 10. FAe semisimple near-ring N is the direct sum of the

two-sided ideals A°.

N is the direct sum of minimal modules according to Theorem 3.

Lemma 1 shows that each of these modules is contained in some one

of the A\. Hence A is the sum of the .4?. It follows from Lemma 2 that

this sum is direct.

Theorem 11. The two-sided ideal A°t of the semisimple near-ring N

is simple when it is considered as a near-ring.

If Mi is a right A° module,

MiN = MiiA°x © • • • © A])

= AM? + • • • + AM.°

= AM? C Mi.

This shows that a right .4?-module is also a right A-module. If M is

an A-module, MCsA* is both an A- and an ^Ij-module. Therefore

the minimal .4j-modules are just the minimal A-modules contained

in A®. This fact, combined with Lemma 1, shows that Al as a near-

ring has one and only one type of irreducible -4j-space. A nilpotent

.4?-module would be a nilpotent A-module. Also, a descending chain

of .4?-modules can be regarded as a chain of A-modules. Therefore,

the semi-simplicity of A implies that A° is semisimple. The conclusion

that A°t is a simple near-ring now follows from Theorem 8.

Theorem 12. IfBisa nonzero two-sided ideal of the semisimple near-

ring N,

B = Ah © • • • © An

for suitable ik.

A given nonzero element b of B can be represented uniquely in the

form

*■«*+•" + <*iP

where a¡k is in A]k and a^^O. Because A\ is simple, it can have no

nonzero annihilators when considered as -4^-space. Therefore A°kajk

?í0. Since



B D A)k(ah + • • • + aip) = Atfit * O, A¡k (~\ B * 0.

Since A°]kr\B is a two-sided ideal of A,

o oAikr\B = AJk.

Therefore

b d A°h e • • • e a)v.

Il ik runs through all the different indices jk which occur for some b

in B, then

B = Ah ® • • • ® Au.

Corollary. If N=Bi® • • • ®Bt where the Bi are indecomposable

two-sided ideals of the semisimple near-ring N, then t = s and after re-

arrangement Bi=Af.

Theorem 13. The semisimple near-ring N is completely determined

up to an isomorphism by the simple near-rings A°.

Theorem 10 states that N=A°t® • • • ®A°. Therefore the addi-

tion in A is determined by the addition in the A*}. It only remains

to be shown that the multiplication in N is determined by the

multiplication in the A°t. This will be proved as soon as it is shown

that

(«i + • • • + a,)(a{ +•••+««')«■ ami + ami + • • • + a,a'.

where a,- and a/ are in A°. Because the A° are right ideals, and

A°i(^( ZA°i) - 0 (Lemma 2),\ Mi /

an argument analogous to the one used in the proof of Theorem 4 to

show that

(«!+•••+ c,+i)r = ep-i + • • • + e,+ir

proves that

(fll+... +a,)(a{ + ... +a¡)

= «i(ai + •••+«.') + ••• + a.(a{ + • • • + a.').

Use of the left distributive law gives

(ai + • • • + a,)(a{ + •••+«.')

= ami + • • • + ami + • • • + a,al + • • • + a.a!.



But by Lemma 2, a,a/' =0 for *Vj.

Theorem 14. // Ai, A2, • • • , and N, are simple near-rings, and N

is defined to be the set of s-tuples

(»i, •••,».)» »¿ E Ni,

then A is a semisimple near-ring if the s-tuples are added and multiplied

componentwise. If the types of irreducible spaces are suitably labeled,

Ai is composed of the s-tuples

(0, • • • , 0, »,-, 0, • • • , 0)

where n¡ is in A,-.

It is clear from the definition of A that A is a near-ring. If M is a

right A-module with elements

m = (»ti, m2, • • • , m,),

the set Mi of all elements »i¿ of Ni which occur as ith components of

some element of M form a right A,-module. Since A contains all

elements of the form

(0, • • • , 0, m, 0, • • • , 0),

it follows that M contains the A-module (AÍ,A<) * which consists of

all elements of the form

(0, • • • , 0, mim, 0, • • • , 0).

If Mi5¿0, Mí = Míx® ■ ■ • ®Muk where each Afy is a minimal right

A-module. Because M a is minimal, AfyAfy«»Afy. Hence AfyA\-

= M a and Af <A< = Mi. Therefore M D M t where M? is the A-module

with elements

m? = (0, • ■ • , 0, mi, 0, • • • , 0).

Since MfC\ Ej> < Mf = 0 and each M,- is a normal subgroup of M,

M=Mx*@ ■ ■ ■ ®M*. If M2>M'DM"D • • • is a descending chain

of A-modules, it gives rise to corresponding chains A/.OAf/D • • •

of A,-modules. By the descending chain condition for right Ai-

modules, each of these chains must stop properly descending after a

finite number of terms. As soon as all of these chains have stopped de-

creasing, so also have the chains MfZ)M'*Z) ■ • • . The formula

Af=Af!*© • • • © M,* now shows that the chain AOAf'DAf'D • • •also stops decreasing. This proves that the right A-modules satisfy

the descending chain condition. If M isa nilpotent A-module, each of

the Ai-modules Mi is nilpotent. This implies that each Af, = 0



(t = l, • • • , s). Hence M=0. This shows that A is semisimple. The

A-module A,* is an indecomposable two-sided ideal of A. Since A

= A*© • • • ©A,*, it follows from the corollary to Theorem 12 that,

with suitable labeling, A° = N?.

References

1. L. E. Dickson, On finite algebras, Nachr. Ges. Wiss. Göttingen (1905) pp. 358-

393.2. H. Zassenhaus, Über endliche Fastkörper, Abh. Math. Sem. Hamburgischen

Univ. vol. 11 (1936) pp. 187-220.3. F. Kaischeuer, Die Bestimmung aller stetigen Fastkörper, über dem Körper der

reellen Zahlen als Grundkörper, Abh. Math. Sem. Hamburgischen Univ. vol. 13 (1940)

pp. 413-435.4. K. Reidemeister, Grundlagen der Geometrie, Berlin, 1930, pp. 110-122.

Princeton University


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