1.
Let
R
be a ring. Recall that its center, denoted
Z
(
R
), is by definition the set of
elements
r
∈
R
such that
r
·
r
0
=
r
0
·
r
for all
r
0
∈
R
.
(a) Let
Z
→
R
be the canonical homomorphism. Show that its image is contained
in the center of
R
.
(b) Let
R
= Mat
n,n
. Show that its center consists of scalar matrices (i.e., multiples
of the identity matrix).
2.
Let
V
be a vector space over
k
, and recall that
T
(
V
) =
⊕
i
≥
0
T
i
(
V
), where
T
i
(
V
) =
V
⊗
...
⊗
V
|
{z
}
i
for
i
≥
1 and
T
0
(
V
) =
k
.
(a

Question

1.
Let
R
be a ring. Recall that its center, denoted
Z
(
R
), is by definition the set of
elements
r
∈
R
such that
r
·
r
0
=
r
0
·
r
for all
r
0
∈
R
.
(a) Let
Z
→
R
be the canonical homomorphism. Show that its image is contained
in the center of
R
.
(b) Let
R
= Mat
n,n
. Show that its center consists of scalar matrices (i.e., multiples
of the identity matrix).
2.
Let
V
be a vector space over
k
, and recall that
T
(
V
) =
⊕
i
≥
0
T
i
(
V
), where
T
i
(
V
) =
V
⊗
...
⊗
V
|
{z
}
i
for
i
≥
1 and
T
0
(
V
) =
k
.
(a