Constructing a graded ring from abelian groups

Constructing a graded ring from abelian groups - defining the unit.

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On page 631 of Lang's Algebra, he gives a construciton

Let $G$ a commutative monoid. Suppose for each $r,s in G$, we have abelian groups $A_r$, and maps $A_r times A_s rightarrow A_r+s$, $A_0$ is a commutative ring and that the composition of these maps is associative and $A_0$-billinear. Then the direct sum
$$A:=bigoplus_r in G A_r$$
is a ring, with product
$$ Big(sum x_r Big) Big( sum y_s Big) = sum_t sum _r+s=t x_ry_s. $$

I don't understand what is means to be an $A_0$-billinear map. I suppose it means
$$ A_0 times A_r rightarrow A_r text is mathbbZ text-billinear.$$

I would like to conclude that $1 in A_0$ is the unit in this new ring $A$. But from the conditions, I could not deduce $1 cdot a = a$ (enough for $a in A_r$, some $r in G$).

Billinearity only gives: $2 cdot a = 2(1cdot a) = 1 cdot 2a.$

edited Sep 7 at 11:10

asked Sep 7 at 7:47

Cyryl L.

1,8632821

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On page 631 of Lang's Algebra, he gives a construciton

Let $G$ a commutative monoid. Suppose for each $r,s in G$, we have abelian groups $A_r$, and maps $A_r times A_s rightarrow A_r+s$, $A_0$ is a commutative ring and that the composition of these maps is associative and $A_0$-billinear. Then the direct sum
$$A:=bigoplus_r in G A_r$$
is a ring, with product
$$ Big(sum x_r Big) Big( sum y_s Big) = sum_t sum _r+s=t x_ry_s. $$

I don't understand what is means to be an $A_0$-billinear map. I suppose it means
$$ A_0 times A_r rightarrow A_r text is mathbbZ text-billinear.$$

I would like to conclude that $1 in A_0$ is the unit in this new ring $A$. But from the conditions, I could not deduce $1 cdot a = a$ (enough for $a in A_r$, some $r in G$).

Billinearity only gives: $2 cdot a = 2(1cdot a) = 1 cdot 2a.$

edited Sep 7 at 11:10

asked Sep 7 at 7:47

Cyryl L.

1,8632821

add a commentÂ |Â

up vote
0
down vote

favorite

On page 631 of Lang's Algebra, he gives a construciton

Let $G$ a commutative monoid. Suppose for each $r,s in G$, we have abelian groups $A_r$, and maps $A_r times A_s rightarrow A_r+s$, $A_0$ is a commutative ring and that the composition of these maps is associative and $A_0$-billinear. Then the direct sum
$$A:=bigoplus_r in G A_r$$
is a ring, with product
$$ Big(sum x_r Big) Big( sum y_s Big) = sum_t sum _r+s=t x_ry_s. $$

I don't understand what is means to be an $A_0$-billinear map. I suppose it means
$$ A_0 times A_r rightarrow A_r text is mathbbZ text-billinear.$$

I would like to conclude that $1 in A_0$ is the unit in this new ring $A$. But from the conditions, I could not deduce $1 cdot a = a$ (enough for $a in A_r$, some $r in G$).

Billinearity only gives: $2 cdot a = 2(1cdot a) = 1 cdot 2a.$

edited Sep 7 at 11:10

asked Sep 7 at 7:47

Cyryl L.

1,8632821

On page 631 of Lang's Algebra, he gives a construciton

Let $G$ a commutative monoid. Suppose for each $r,s in G$, we have abelian groups $A_r$, and maps $A_r times A_s rightarrow A_r+s$, $A_0$ is a commutative ring and that the composition of these maps is associative and $A_0$-billinear. Then the direct sum
$$A:=bigoplus_r in G A_r$$
is a ring, with product
$$ Big(sum x_r Big) Big( sum y_s Big) = sum_t sum _r+s=t x_ry_s. $$

I don't understand what is means to be an $A_0$-billinear map. I suppose it means
$$ A_0 times A_r rightarrow A_r text is mathbbZ text-billinear.$$

I would like to conclude that $1 in A_0$ is the unit in this new ring $A$. But from the conditions, I could not deduce $1 cdot a = a$ (enough for $a in A_r$, some $r in G$).

Billinearity only gives: $2 cdot a = 2(1cdot a) = 1 cdot 2a.$

abstract-algebra differential-geometry modules graded-rings

edited Sep 7 at 11:10

asked Sep 7 at 7:47

Cyryl L.

1,8632821

edited Sep 7 at 11:10

asked Sep 7 at 7:47

Cyryl L.

1,8632821

edited Sep 7 at 11:10

asked Sep 7 at 7:47

Cyryl L.

1,8632821

asked Sep 7 at 7:47

Cyryl L.

1,8632821

asked Sep 7 at 7:47

Cyryl L.

1,8632821

add a commentÂ |Â

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