Generalized higher derivations on algebras
We study the structure of generalized higher derivations on an algebra ${\scr A}$ and show that there exists a one-to-one correspondence between the set of all generalized higher derivations $\{ G_k\}^n_{k =0}$ on ${\scr A}$ with $G_0 = I$ and the set of all sequences $\{ g_k\}^n_{k = 0}$ of gene...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507654140461056 |
|---|---|
| author | Fošner, A. Фоснер, А. |
| author_facet | Fošner, A. Фоснер, А. |
| author_sort | Fošner, A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-09T06:03:14Z |
| description | We study the structure of generalized higher derivations on an algebra ${\scr A}$ and show that there exists a one-to-one
correspondence between the set of all generalized higher derivations $\{ G_k\}^n_{k =0}$ on ${\scr A}$ with $G_0 = I$ and the set of all
sequences $\{ g_k\}^n_{k = 0}$ of generalized derivations on ${\scr A}$ with $g_0 = 0$. |
| first_indexed | 2026-03-24T02:12:45Z |
| format | Article |
| fulltext |
UDC 512.5
A. Fošner (Univ. Primorska, Slovenia)
GENERALIZED HIGHER DERIVATIONS ON ALGEBRAS
УЗАГАЛЬНЕНI ВИЩI ПОХIДНI НА АЛГЕБРАХ
We study the structure of generalized higher derivations on an algebra \scrA and show that there exists a one-to-one
correspondence between the set of all generalized higher derivations \{ Gk\} nk=0 on \scrA with G0 = I and the set of all
sequences \{ gk\} nk=0 of generalized derivations on \scrA with g0 = 0.
Вивчено структуру узагальнених вищих похiдних на алгебрi \scrA та показано, що iснує взаємно однозначна вiдповiд-
нiсть мiж множиною всiх узагальнених вищих похiдних \{ Gk\} nk=0 на \scrA з G0 = I та множиною всiх послiдовностей
\{ gk\} nk=0 узагальнених похiдних на \scrA з g0 = 0.
1. Introduction. Let \scrA be an associative algebra and I the identity map on \scrA . A linear mapping
d : \scrA \rightarrow \scrA is a derivation if
d(xy) = d(x)y + xd(y), x, y \in \scrA .
A derivation d is called inner if there exists a fixed element a \in \scrA such that d(x) = [a, x] = ax - xa
for all x \in \scrA . Now, let n \in \BbbN \cup \{ 0,\infty \} and let \{ Dk : \scrA \rightarrow \scrA | k = 0, 1, . . . , n\} be a family of
linear mappings with D0 = I. The sequence D =
\bigl\{
Dk
\bigr\} n
k=0
is called a higher derivation of rank n
if
Dk(xy) =
k\sum
i=0
Di(x)Dk - i(y), x, y \in \scrA , k = 0, 1, . . . , n. (1)
Obviously, D1 satisfies D1(xy) = D0(x)D1(y)+D1(x)D0(y) = xD1(y)+D1(x)y for all x, y \in \scrA .
Thus, D1 is a derivation on \scrA . If d : \scrA \rightarrow \scrA is a derivation, then D =
\biggl\{
dk
k!
\biggr\} n
k=0
is a standard
example of a higher derivation of rank n. This kind of higher derivation is called an ordinary higher
derivation. Note also that this is not the only example of higher derivations.
Higher derivations have been considered for the first time by Hasse and Schmidt [4] and inde-
pendently by Teichm\"\mathrm{u}ller [10], being difficult to affirm with certainty which of them was the pioneer
in this study. The concept of higher derivations was originally introduced in order to remove some
of the anomalies in the calculus of derivations on fields of characteristic p \not = 0 (cf. [3, 6]). Never-
theless, higher derivations have important applications in characteristic zero situations, as shown by
Heerema [5]. In a celebrated paper [5], Heerema showed a review of the theory of higher derivations
on fields which form a background for the study of the uses of higher derivations in automorphism
theory of complete local rings. For more information about higher derivations, their chronological
development, importance and applications we refer the reader to [1] and [2].
A linear mapping g : \scrA \rightarrow \scrA is a generalized derivation if there exists a derivation d : \scrA \rightarrow \scrA
such that
g(xy) = g(x)y + xd(y), x, y \in \scrA .
c\bigcirc A. FOŠNER, 2017
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 10 1429
1430 A. FOŠNER
In this case we sometimes call g a generalized d-derivation (i.e., g is a generalized derivation
associated with the derivation d). If \scrA has a unit 1 \in \scrA , then we can write the above equality
as g(xy) = g(x)y + xg(y) - xg(1)y, x, y \in \scrA . A generalized derivation g is called inner if there
exist elements a, b \in \scrA such that g(x) = ax + xb for all x \in \scrA . The sequence of linear mappings
G = \{ Gk\} nk=0 is called a generalized higher derivation of rank n if there exists a higher derivation
D = \{ Dk\} nk=0 such that
Gk(xy) =
k\sum
i=0
Gi(x)Dk - i(y), x, y \in \scrA , k = 0, 1, . . . , n. (2)
In this case G is a generalized higher derivation associated with the higher derivation D. Thus, we
sometimes call G simply a generalized higher D-derivation. Of course, if D = G, then G is just a
higher derivation.
As far as we are aware, the above definition was first introduced by Ribenboim in [9]. Ribenboim
gave some properties of higher derivations of modules. His higher derivation f from an \scrA -module
\scrM to \scrM is defined by using a higher derivation D = \{ Dk\} nk=0 : \scrA \rightarrow \scrA . In 2000, Nakajima [8]
(Definition 2.1) defined generalized higher derivations from an R-algebra \scrA (here R is a commutative
ring) to an \scrA /R-bimodule \scrM and gave some of their categorical properties. He also treated
generalized higher Jordan and Lie derivations. For more results on generalized higher derivations we
refer the reader to the survey [2].
Let us continue with a standard example of generalized higher derivations which is, of course,
not the only example of generalized higher derivations (see Section 2).
Example 1. Let g : \scrA \rightarrow \scrA be a generalized d-derivation. Then it is easy to see that G =
=
\biggl\{
gk
k!
\biggr\} n
k=0
is a generalized higher derivation associated with the ordinary higher derivation D =
=
\biggl\{
dk
k!
\biggr\} n
k=0
.
The concept of generalized higher derivations is rather new and relatively few results have been
obtained so far concerning it. This motivated us to study the structure of generalized higher derivations
and to generalize some known results. We show that there exists a one to one correspondence between
the set of all generalized higher derivations \{ Gk\} nk=0 on \scrA with G0 = I and the set of all sequences
\{ gk\} nk=0 of generalized derivations on \scrA with g0 = 0. The importance of our work is to transfer
the problems such as innerness and automatic continuity of generalized higher derivations into the
same problems concerning generalized derivations. Namely, we will prove that each component of a
generalized higher derivation is a combination of compositions of generalized derivations.
2. Generalized higher derivations. Throughout the paper, \scrA denotes an associative algebra
over a field of characteristic zero. Recently, Mirzavaziri [7] characterized all higher derivations on
an algebra \scrA in terms of derivations on \scrA . In particular, he proved that each component of a higher
derivation is a combination of compositions of derivations. The natural question here is whether the
analogue result holds true for generalized higher derivations. Theorem 1 answers this question in the
affirmative.
Theorem 1. Let G = \{ Gk\} nk=0 be a generalized higher derivation on \scrA with G0 = I. Then
there exists a sequence \{ gk\} nk=1 of generalized derivations on \scrA such that
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 10
GENERALIZED HIGHER DERIVATIONS ON ALGEBRAS 1431
Gk =
k\sum
i=1
\left( \sum
\sum i
j=1 rj=k
\left( i\prod
j=1
1
rj + . . .+ ri
\right) gr1 . . . gri
\right) , (3)
where the inner summation is taken over all positive integers rj such that
\sum i
j=1
rj = k.
Below we use the characterization of higher derivations proved by Mirzavaziri [7]. Recall that
higher derivations in the sense of Mirzavaziri are infinite sequences of linear maps satisfying certain
conditions while (generalized) higher derivations in this paper can be finite or infinite sequences of
linear maps satisfying certain conditions. Nevertheless, it is easy to see (following the proofs in [7])
that the results in [7] hold true in the finite case as well.
Proof of Theorem 1. Suppose that G = \{ Gk\} nk=0 is a generalized higher derivation associated
with the higher derivation D = \{ Dk\} nk=0. Then, by Theorem 2.3 in [7], there exists a sequence
\{ dk\} nk=1 of derivations on \scrA such that
Dk =
k\sum
i=1
\left( \sum
\sum i
j=1 rj=k
\left( i\prod
j=1
1
rj + . . .+ ri
\right) dr1 . . . dri
\right) . (4)
Here, always, the inner summation is taken over all positive integers rj such that
\sum i
j=1
rj = k. We
divide the proof into two steps.
Step 1. Using the induction on k, we first show that there exists a sequence \{ gk\} nk=1 of generalized
dk -derivations on \scrA such that
(k + 1)Gk+1 =
k\sum
i=0
gi+1Gk - i (5)
for each k = 0, 1, . . . , n - 1.
First of all, it is easy to see that G1 is a generalized derivation. Namely, for all x, y \in \scrA , we
have
G1(xy) = G0(x)D1(y) +G1(x)D0(y) = xD1(y) +G1(x)y.
Recall that, according to (4), D1 = d1 is a derivation and, thus, G1 = g1 is a generalized d1-
derivation. In particular, we have proved (5) for k = 0.
Now, suppose that k \geq 1 and that there exist appropriate generalized derivations g1, . . . , gk on
\scrA such that (5) holds true. Let us define a mapping gk+1 : \scrA \rightarrow \scrA by
gk+1 = (k + 1)Gk+1 -
k - 1\sum
i=0
gi+1Gk - i.
Of course, gk+1 is linear. Now, let x, y \in \scrA be arbitrary elements. Then
gk+1(xy) = (k + 1)Gk+1(xy) -
k - 1\sum
i=0
gi+1Gk - i(xy) =
= (k + 1)
k+1\sum
i=0
Gi(x)Dk+1 - i(y) -
k - 1\sum
i=0
gi+1
\left( k - i\sum
j=0
Gj(x)Dk - i - j(y)
\right) .
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 10
1432 A. FOŠNER
Furthermore, since g1, . . . , gk are generalized derivations associated with derivations d1, . . . , dk, we
have
gk+1(xy) =
k+1\sum
i=0
iGi(x)Dk+1 - i(y) +
k+1\sum
i=0
(k + 1 - i)Gi(x)Dk+1 - i(y) =
-
k - 1\sum
i=0
k - i\sum
j=0
\bigl(
gi+1
\bigl(
Gj(x)
\bigr)
Dk - i - j(y) +Gj(x)di+1
\bigl(
Dk - i - j(y)
\bigr) \bigr)
.
Let us divide the above sum into two parts
S1 =
k+1\sum
i=0
iGi(x)Dk+1 - i(y) -
k - 1\sum
i=0
k - i\sum
j=0
gi+1
\bigl(
Gj(x)
\bigr)
Dk - i - j(y),
S2 =
k+1\sum
i=0
(k + 1 - i)Gi(x)Dk+1 - i(y) -
k - 1\sum
i=0
k - i\sum
j=0
Gj(x)di+1
\bigl(
Dk - i - j(y)
\bigr)
.
Of course, gk+1(xy) = S1 + S2.
Let us first compute S1. Recall that in the summation
\sum k - 1
i=0
\sum k - i
j=0
we have 0 \leq i+ j \leq k and
i \not = k. Thus, if we write l = i+ j, we get
S1 =
k+1\sum
i=0
iGi(x)Dk+1 - i(y) -
k\sum
l=0
\sum
0\leq i\leq l,i \not =k
gi+1
\bigl(
Gl - i(x)
\bigr)
Dk - l(y) =
=
k+1\sum
i=0
iGi(x)Dk+1 - i(y) -
k - 1\sum
l=0
l\sum
i=0
gi+1
\bigl(
Gl - i(x)
\bigr)
Dk - l(y) -
-
k - 1\sum
i=0
gi+1
\bigl(
Gk - i(x)
\bigr)
y.
Therefore, writing l + 1 instead of i in the first summation, we obtain
S1 =
k\sum
l=0
(l + 1)Gl+1(x)Dk - l(y) -
k - 1\sum
l=0
l\sum
i=0
gi+1
\bigl(
Gl - i(x)
\bigr)
Dk - l(y) -
-
k - 1\sum
i=0
gi+1
\bigl(
Gk - i(x)
\bigr)
y =
=
k - 1\sum
l=0
\Biggl(
(l + 1)Gl+1(x)Dk - l(y) -
l\sum
i=0
gi+1
\bigl(
Gl - i(x)
\bigr)
Dk - l(y)
\Biggr)
+
+(k + 1)Gk+1(x)y -
k - 1\sum
i=0
gi+1
\bigl(
Gk - i(x)
\bigr)
y.
By our assumption,
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 10
GENERALIZED HIGHER DERIVATIONS ON ALGEBRAS 1433
(l + 1)Gl+1(x) =
l\sum
i=0
gi+1
\bigl(
Gl - i(x)
\bigr)
for l = 0, 1, . . . , k - 1. Thus,
S1 =
\Biggl(
(k + 1)Gk+1(x) -
k - 1\sum
i=0
gi+1
\bigl(
Gk - i(x)
\bigr) \Biggr)
y = gk+1(x)y.
Using the fact that dk+1 = (k + 1)Dk+1 -
\sum k - 1
i=0
di+1Dk - i, we can similarly deduce that
S2 = xdk+1(y).
Therefore,
gk+1(xy) = gk+1(x)y + xdk+1(y).
Since x and y were arbitrary elements from \scrA , this yields that gk+1 is a generalized dk+1-derivation,
as desired.
Step 2. Suppose that Gk is of the form (3) for k \geq 1. If we denote
\alpha r1,...,ri =
i\prod
j=1
1
rj + . . .+ ri
,
then we have
(k + 1)Gk+1 = (k + 1)
k+1\sum
i=1
\left( \sum
\sum i
j=1 rj=k+1
\alpha r1,...,rigr1 . . . gri
\right) =
=
k+1\sum
i=2
\left( \sum
\sum i
j=1 rj=k+1
(k + 1)\alpha r1,...,rigr1 . . . gri
\right) + gk+1.
Note that (k + 1)\alpha r1,...,ri = \alpha r2,...,ri since r1 + . . .+ ri = k + 1. Therefore,
(k + 1)Gk+1 =
k+1\sum
i=2
\left( k+2 - i\sum
r1=1
gr1
\sum
\sum i
j=2 rj=k+1 - r1
\alpha r2,...,rigr2 . . . gri
\right) + gk+1 =
=
k\sum
r1=1
gr1
k - (r1 - 1)\sum
i=1
\left( \sum
\sum i
j=2 rj=k - (r1 - 1)
\alpha r2,...,rigr2 . . . gri
\right) + gk+1 =
=
k\sum
r1=1
gr1Gk - (r1 - 1) + gk+1.
If we write r1 = i, we get
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 10
1434 A. FOŠNER
(k + 1)Gk+1 =
k - 1\sum
i=0
gi+1Gk - i + gk+1 =
k\sum
i=0
gi+1Gk - i.
Namely, gk+1 = gk+1I = gk+1G0. Consequently, G1, . . . , Gn satisfy the recursive relation (5).
Since the solution of the recursive relation in Step 1 is unique, this completes the proof.
Theorem 2. Let \Phi be the set of all generalized higher derivations \{ Gk\} nk=0 on \scrA with G0 = I
and \Gamma the set of all sequences \{ gk\} nk=0 of generalized derivations on \scrA with g0 = 0. Then there is
a one-to-one correspondence between \Phi and \Gamma .
Proof. Let G = \{ Gk\} nk=0 \in \Phi be a generalized higher derivation on \scrA . Then, by Theorem 1,
there exists a sequence g = \{ gk\} nk=0 of generalized derivations on \scrA with g0 = 0 such that (5) holds
true for each k = 0, 1, . . . , n - 1. Thus, we can define a mapping \varphi : \Phi \rightarrow \Gamma by
\{ Gk\} nk=0 = G
\varphi - \rightarrow g = \{ gk\} nk=0 .
Note that a map \varphi is injective since the solution of the recursive relation (5) is unique.
It remains to prove that \varphi is surjective. For a given sequence g = \{ gk\} nk=0 \in \Gamma of generalized
dk -derivations on \scrA with g0 = 0, one can define mappings G0, . . . , Gn with G0 = I and
(k + 1)Gk+1 =
k\sum
i=0
gi+1Gk - i
for k = 0, 1, . . . , n - 1. We have to prove that G = \{ Gk\} nk=0 is a generalized higher derivation.
Obviously, Gk are linear mappings on \scrA . Furthermore, we know that there exists a higher derivation
D = \{ Dk\} nk=0 such that
(k + 1)Dk+1 =
k\sum
i=0
di+1Dk - i
for k = 0, 1, . . . , n - 1.
Let x, y \in \scrA . Using the induction on k, we show that
Gk(xy) =
k\sum
i=0
Gi(x)Dk - i(y), k = 0, 1, . . . , n. (6)
For k = 0, we have G0(xy) = xy = G0(x)D0(y) and, for k = 1, we have
G1(xy) = g1G0(xy) = g1(xy) = g1(x)y + xd1(y) =
= G1(x)D0(y) +G0(x)D1(y).
Assume now that Gl(xy) =
\sum l
i=0
Gi(x)Dl - i(y) for l = 0, 1, . . . , k. Then
(k + 1)Gk+1(xy) =
k\sum
i=0
gi+1Gk - i(xy) =
=
k\sum
i=0
gi+1
k - i\sum
j=0
Gj(x)Dk - i - j(y) =
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 10
GENERALIZED HIGHER DERIVATIONS ON ALGEBRAS 1435
=
k\sum
i=0
k - i\sum
j=0
\Bigl(
gi+1Gj(x)Dk - i - j(y) +Gj(x)di+1Dk - i - j(y)
\Bigr)
.
Therefore,
(k + 1)Gk+1(xy) =
k\sum
j=0
\Biggl(
k - j\sum
i=0
gi+1Gk - j - i(x)
\Biggr)
Dj(y)+
+
k\sum
j=0
Gj(x)
k - j\sum
i=0
di+1Dk - j - i(y) =
k\sum
j=0
(k - j + 1)Gk - j+1(x)Dj(y)+
+
k\sum
j=0
Gj(x)(k - j + 1)Dk - j+1(y) =
k+1\sum
j=1
jGj(x)Dk+1 - j(y)+
+
k\sum
j=0
(k - j + 1)Gj(x)Dk+1 - j(y) = (k + 1)
k+1\sum
j=0
Gj(x)Dk+1 - j(y).
Thus, we have proved (6).
Theorem 2 is proved.
3. Additional remarks. In this last section of the paper we write some additional observations
about our results. The first remark is connected with Example 1.
Remark 1. It is easy to see that a generalized higher derivation G = \{ Gk\} nk=0 = \varphi - 1
\bigl(
\{ gk\} nk=0
\bigr)
is an ordinary generalized higher derivation if and only if gk = 0 for k \geq 2. In this case
Gk =
gk1
k!
, k = 0, 1, . . . , n.
Remark 2. In our definition of higher derivations we assumed that D0 is the identity map on \scrA .
But if D0 is any automorphism of \scrA and D =
\bigl\{
Dk
\bigr\} n
k=0
a sequence of linear mappings with the
property (1), then it is easy to see that D1 is a (D0, D0)-derivation, i.e.,
D1(xy) = D1(x)D0(y) +D0(x)D1(y), x, y \in \scrA ,
and (following the proof of Proposition 2.1 in [7]), we can deduce that the sequence \{ Dk\} nk=0
corresponds to the sequence \{ dk\} nk=0 of (D0, D0)-derivations on \scrA with d0 = 0.
Similarly, we can define a generalized higher derivation as a sequence of linear mappings G =
= \{ Gk\} nk=0, where G0 is any automorphism of \scrA , such that (2) holds true for all x, y \in \scrA ,
k = 0, 1, . . . , n, and some higher derivation D = \{ Dk\} nk=0 with D0 = G0. Suppose that \{ Dk\} nk=0
corresponds to the sequence \{ dk\} nk=0 of (G0, G0)-derivations on \scrA with d0 = 0. For all k =
= 0, 1, . . . , n, let us denote \^Gk = GkG
- 1
0 , \^Dk = DkG
- 1
0 , and \^dk = dkG
- 1
0 . Then it is easy
to see that \^G = \{ \^Gk\} nk=0 is a generalized higher derivation associated with a higher derivation
\^D = \{ \^Dk\} nk=0. Moreover, \^G0 = \^D0 = I. Thus, there exists a sequence \{ \^gk\} nk=1 of generalized
\^dk -derivations such that (3) holds true for k = 1, . . . , n. Now, one can easily conclude that the
sequence \{ Gk\} nk=0 corresponds to the sequence \{ \^gkG0\} nk=0 of generalized (G0, G0)-derivations on
\scrA associated with (G0, G0)-derivations \{ \^dkG0\} nk=0 = \{ dk\} nk=0. Here, of course, \^g0 = 0 = \^d0.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 10
1436 A. FOŠNER
Remark 3. Similarly as higher derivations and generalized higher derivations, we can define
higher Jordan (Lie) derivations and generalized higher Jordan (Lie) derivations. Moreover, following
the proofs in Section 2, we can deduce that there is a one to one correspondence between the set of
all (generalized) higher Jordan derivations \{ Jk\} nk=0 on \scrA with J0 = I and the set of all sequences
\{ jk\} nk=0 of (generalized) Jordan derivations on \scrA with j0 = 0. Similarly, there is a one-to-one
correspondence between the set of all (generalized) higher Lie derivations \{ Lk\} nk=0 on \scrA with
L0 = I and the set of all sequences \{ lk\} nk=0 of (generalized) Lie derivations on \scrA with l0 = 0.
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Received 12.01.16
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 10
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| id | umjimathkievua-article-1792 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:12:45Z |
| publishDate | 2017 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/40/d36204ba88828a6b2164bb56cd786140.pdf |
| spelling | umjimathkievua-article-17922019-12-09T06:03:14Z Generalized higher derivations on algebras Узагальненi вищi похiднi на алгебрах Fošner, A. Фоснер, А. We study the structure of generalized higher derivations on an algebra ${\scr A}$ and show that there exists a one-to-one correspondence between the set of all generalized higher derivations $\{ G_k\}^n_{k =0}$ on ${\scr A}$ with $G_0 = I$ and the set of all sequences $\{ g_k\}^n_{k = 0}$ of generalized derivations on ${\scr A}$ with $g_0 = 0$. Вивчено структуру узагальнених вищих похiдних на алгебрi ${\scr A}$ та показано, що iснує взаємно однозначна вiдповiднiсть мiж множиною всiх узагальнених вищих похiдних $\{ G_k\}^n_{k =0}$ на $\scr A$ з $G_0 = I$ та множиною всiх послiдовностей $\{ g_k\}^n_{k = 0}$ узагальнених похiдних на $\scr A$ з $g_0 = 0$. Institute of Mathematics, NAS of Ukraine 2017-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1792 Ukrains’kyi Matematychnyi Zhurnal; Vol. 69 No. 10 (2017); 1429-1436 Український математичний журнал; Том 69 № 10 (2017); 1429-1436 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1792/774 Copyright (c) 2017 Fošner A. |
| spellingShingle | Fošner, A. Фоснер, А. Generalized higher derivations on algebras |
| title | Generalized higher derivations on algebras |
| title_alt | Узагальненi вищi похiднi на алгебрах |
| title_full | Generalized higher derivations on algebras |
| title_fullStr | Generalized higher derivations on algebras |
| title_full_unstemmed | Generalized higher derivations on algebras |
| title_short | Generalized higher derivations on algebras |
| title_sort | generalized higher derivations on algebras |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1792 |
| work_keys_str_mv | AT fosnera generalizedhigherderivationsonalgebras AT fosnera generalizedhigherderivationsonalgebras AT fosnera uzagalʹneniviŝipohidninaalgebrah AT fosnera uzagalʹneniviŝipohidninaalgebrah |