奈良教育大学学術リポジトリNEAR
Some remarks on purely inseparable algebras
著者 KIKUCHI Teppei
journal or
publication title
奈良教育大学紀要. 自然科学
volume 25
number 2
page range 1‑4
year 1976‑12‑25
URL http://hdl.handle.net/10105/2554
Some remarks on purely inseparable algebras
Teppei KlKUCHl
(Department of Mathematics, Nara University of Education, Nara, Japan) (Received April 28, 1976)
M. E. Sweedler defined in his rescent paper [1] the notion of a purely insepa- rable algebra, and studied the structure of such an algebra over a field. In this short paper, we treat some related topics.
We assume all rings in consideration have identities. J(A) denotes always theJacobson radical of a ring A. The unadorned symbol "®" always means tensoring with respect to k.
1. Let A be an algebra over a commutative ring k. We consider the so called enveloping algebra A®kA0 of A, where A" denotes the opposite algebra
to A. By Ya.A^hA0 >A wealways mean a map defined by 7/t(2«!®#*°)
i
=11aibi.
iWeregard A as a left A®A"-module by setting {a®b°)c=acb.
Then ja is an A®A°-module map. However yA is not always an algebra map, it is an algebra map if and only if A is commutative. Now, after Sweedler's
[1], we call A is a purely inseparable ^-algebra if jA:A®A°->A gives an A®^4°-projective cover of A. In other words, a ^-algebra A is purely insepa- rableifand onlyifKer(yA:A®A° *A) is a small ideal, that is, Ker(yA)ci
J(A®A°).
2. Lemma 1. LetA andBbe two rings andB be a leftA-module. Let
f:B >A be anA-homomorphism such that /(1B)=\a and a-b-(a-lB)b for
every a^A andb^B. IftheKer(f) is a two-sided ideal of B, then f is a
ring homomorphism.
Proof. We shall show that f(ab)=f(a)f(b) for any two elements a and
b in B. Ifweset a=f{a), then f{a-a-lB)=f{a)-afilB)-a-a-Q.
Hence a~a-lB^Ker(f). Therefore ab-a-b=(a-a-lB)b^Ker(f), because
Ker(f) is an ideal of B. Consequently we have f{ab)-f(a-b)=af(b)=
f(a)f(b). Q.E. D.
Proposition 2. Let A be a purely inseparable algebra over a commutative
ring k , and B be a separablek-algebra contained in A. (\B need not a-
gree with 1a.) Then B is a commutative ring. Moreover, if 1B=1a, B is
contained in the center Z(A) of A and A is apurely inseparable B-algebra.
Proof. Let A:B »A be the inclusion map. In the commutative diagram
1
Teppei Kikuchi
B®B°
B
A®A°
A®A°-
7b
Ta -^A
it holds that Ker(yB)c:Ker(A®A0) because B is a separable ^-algebra, (cf.
Sweedier [1] Prop. 6.(a). This proposition holds without assumption Ib - Ia.) On the other hand, we have Ker(yB)^>Ker(A®A°) because A is injective.
Hence Ker{yB)=Ker(A®A°). Since A®A" is an algebra map, it follows that
Ker{AB) is a two-sidedidealof B®B°. Consequently yB:B®B° >Bis an
algebra map by Lemma 1. Therefore B must be commutative.
Next we show that B<^Z(A), provided that 1b = 1a. For an arbitrary ele-
ment b in B, wehave 1®b°-b®l°^Ker(yB)=Ker(A®A°). Hence l®A(b)°
=A(b)®l". This means that l®b°= b®l° holds in A®A0. Consequently, for anyelement a in A, wehave
a®b°=(a®lo)(l®bo)=(a®lo)(b®l°)=ab®l0,
and a®b°=(l®bo)(a®lo)=(b®lo)(a®l°)= ba®l0.
Therefore ab= yA(a®b°)= ba. Thus we have BcZ(A).
The last assertion follows immediately from Sweedler [ll Prop.6.(b).Q.E.D.
?. Now we suppose that k is a field. Sweedler [l] proved the following two propositions.
Proposition A. Let A be an algebraover afield k. If A has GB radi-
cal (that is, for each ^-algebra. B, we have lm(J(A)®B->A®B)<zJ{A®B)), then A is a purely inseparable ^-algebra if and only if J(A) is a nil ideal and A/J(A) is a purely inseparable field extension of k.
Proposition B. Let A be a purely inseparable ^-algebra over a field k and C be a commutative subalgebra of A. Then C is also a purely inseparable k- algebra.
In connection with these propositions, Sweedler set forth the following ques- tions.
(a) Can "GB radical" condition be removed in Proposition A?
(b) Can "commutative" condition be removed in Proposition B?
Relating these questions, we first study the properties of a ^-algebra A such that J{A) is a nil ideal and A/J(A) is a purely inseparable field extension of k. And we shall show, as a consequence, that if the question (a) is affirmative then (b) is also.
4. Let A beanalgebraoverafield k, and suppose that J{A) is a nil
ideal and A/J(A) is a purely inseparable field extension (in the usual sense)
of k.
i ) First we claim that every element of A is either nilpotent or invertible.
For, if a^J(A), a is evidently nilpotent baceuse J(A) is nil. If a&A and a&J(A), setting d-(p(a) theimageof a underthenaturalmap <p:A *A/
J(A), there exists a non nagative integer e such that dpe<^k, where p is the characteristic of k(if k is of characteristic 0, we set pe=1). Set a=dpe.
Then a is a non zero element of k, and ape-a^J(A) because <p(ape-a)=0.
Hence a~l-ape-l^J(A), that is a-l-ape=l modulo J{A). Therefore, by the
definition of Jacobson rabical, a~l-ape is invertible in A. Consequently a is also invertible in A. Thus we have proved that each element of A is either nilpotent or invertible and J(A) is the totality of nilpotent elements of A.
ii) Next let C bea subalgebra ofA. Then ](A)DC is anilidealof C,
hence ](A)(~)C is contained in J{C). On the other hand,since kczC/(J(A)(~)C) CA//(A) and A/J(A) is a purely inseparable field extention of k, C//(A)DC) is also a purely inseparable field extension of k. Consequently /(A)DC is a maximal one-sided ideal of C, hence J(A)C\C contains J(C). Therefore we conclude that J(A)C\C -J(C). Thus we have proved that, for every subalge- bra C of A, J{C) is a nil ideal and C/J(C) is a purely inseparable field ex- tension of k.
iii) Finally let 6 be the family of all purely inseparabele subalgebras of A.
Obviously 6 is non-empty. Let {CJa&i be a linearly ordered sub family of 6. If we set C~ U Ck (set theoretic union of C/s), clearly C is a subalgebra of A.
Hence, by the argument of paragraph ii), J{C) is a nil ideal and ClJ(C) is a purely inseparable field extension of k. Furthermore it holds that J{C) ® C°c/
(C®C°) and C®J{C°)c/(C®C°). For, if £ belongs to /(C)®C°,there
exists a suffix /*eyl such that P<^J{C,)® Cx since /(C)(1G=J(Cx). How- ever, because Ct is a purely inseparable ^-algebra,we know that J{Ci) ® d°c J(Cx® Cx°). (cf. [1], Theorem 12.) Hence j3 lies in /(Cj®C?) andconsequent- ly 1®1°-# is invertible in d® Ca°. Therefore 1®l°-/3 is invertibelein C®
C°. Ontheotherhand /(C)®C° isanidealof C®C°. Thus /9 belongs to
/(C®C°), which means that J(C)® C°C/(C® C°). Similarly we can show
that C®/(C°)c/(C®C°).
These facts show that C is a purely inseparable ^-algebra on account of [l], Theorem 12. Thus we have proved that 6 is inductive, and we can apply the Zorn's lemma.
Summarizing the above arguments, we obtain the following
Proposition 3. Let A be an algebra overafield k such that J(A) is a
nil ideal and A/J(A) is a purely inseparable field extension ofk. Then,{ i )
4 Teppei Kikuchi
each element of A is either nilpotent or invertible and J(A) is the totality of nilpotent elements of A.
(ii) Every subalgebra C of A sctisfies the same conditions with A,
that is, J(C) is a nilideal and C/J(C) is a purely inseparable field exten- sionofk.
(iii) Let B be a purely inseparable k-subalgebra of A. Then thereex-
ists a maximal purely inseparable k-subalgebra of A containing B.
Corollary 4. // the question (a) is affirmative, then the question (b) is also affirmative.
Reference
[1]. Sweedler, M. E. Purely inseparable algebras, J. Algebra 35, 342-355(1975).
