definition of quotient ring of ring by ideal
Topics
About: ring
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Note
Starting Context
- The reader knows a definition of ring.
- The reader knows a definition of ideal of ring.
- The reader knows a definition of quotient group of group by normal subgroup.
Target Context
- The reader will have a definition of quotient ring of ring by ideal.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( R\): \(\in \{\text{ the rings }\}\)
\( I\): \(\in \{\text{ the both-sided ideals of } R\}\)
\(*R / I\): \(= \text{ the additive quotient group }\) as the ring with the multiplication specified below
//
Conditions:
\(\forall [p_1], [p_2] \in R / I ([p_1] [p_2] = [p_1 p_2])\)
//
As \(I\) is an additive normal subgroup, the additive quotient group makes sense. The addition of \(R / I\) is of course the addition of the additive quotient group.
The multiplication is well-defined: for each \([p_1] = [p'_1]\) and \([p_2] = [p'_2]\), there are some \(i_1, i_2 \in I\) such that \(p'_1 = p_1 + i_1\) and \(p'_2 = p_2 + i_2\), and \([p'_1 p'_2] = [(p_1 + i_1) (p_2 + i_2)] = [p_1 p_2 + p_1 i_2 + i_1 p_2 + i_1 i_2] = [p_1 p_2]\), because \(p_1 i_2 + i_1 p_2 + i_1 i_2 \in I\).
2: Natural Language Description
For any ring, \(R\), and its any both-sided ideal, \(I\), the additive quotient group, \(R / I\), as the ring with the multiplication, \([p_1] [p_2] = [p_1 p_2]\)
3: Note
\(R / I\) is indeed a ring: 1) it is an Abelian group under addition: it is the additive quotient group of the Abelian group; 2) it is a monoid under multiplication: the associativity holds, because \(([p_1] [p_2]) [p_3] = [p_1 p_2] [p_3] = [(p_1 p_2) p_3] = [p_1 (p_2 p_3)] = [p_1] [p_2 p_3] = [p_1] ([p_2] [p_3])\); \([1]\) is the identity, because \([1] [p] = [1 p] = [p] = [p 1] = [p] [1]\); 3) multiplication is distributive with respect to addition: \([p_1] ([p_2] + [p_3]) = [p_1] [p_2 + p_3] = [p_1 (p_2 + p_3)] = [p_1 p_2 + p_1 p_3] = [p_1 p_2] + [p_1 p_3] = [p_1] [p_2] + [p_1] [p_3]\) and \(([p_1] + [p_2]) [p_3] = [p_1 + p_2] [p_3] = [(p_1 + p_2) p_3] = [p_1 p_3 + p_2 p_3] = [p_1 p_3] + [p_2 p_3] = [p_1] [p_3] + [p_2] [p_3]\).
