For any (not necessarily commutative) algebra C over a commutative ring k Sweedler defined a cohomology set, denoted here by
H
2
(
C
/
k
)
{\mathcal {H}^2}(C/k)
, which generalizes Amitsur’s second cohomology group
H
2
(
C
/
k
)
{H^2}(C/k)
. In this paper, if I is a nilpotent ideal of C and
C
¯
≡
C
/
I
\bar C\, \equiv \,C/I
is K-projective, a natural bijection
H
2
(
C
/
k
)
→
~
H
2
(
C
¯
/
k
)
{\mathcal {H}^2}(C/k)\tilde \to {\mathcal {H}^2}(\bar C{\text {/}}k)
is established. Also, when
k
⊂
B
k \subset B
are fields and C is a commutative B-algebra, the sequence
{
1
}
→
H
2
(
B
/
k
)
→
l
∗
H
2
(
C
/
k
)
→
r
H
2
(
C
/
B
)
\{ 1\} \to {H^2}(B{\text {/}}k)\xrightarrow {{{l^{\ast }}}}{H^2}(C/k)\xrightarrow {r}{H^2}(C/B)
is shown to be exact if the natural map
C
⊗
k
C
→
C
⊗
B
C
C{ \otimes _k}C \to C{ \otimes _B}C
induces a surjection on units,
l
∗
{l^ {\ast } }
is induced by the inclusion, and r is the “restriction” map.