Ideal

Last updated January 27, 2022

Let $A$ be a commutative ring with unit.

Idea

Given a subset $S\subset A$, the definition of an ideal is precisely the conditions necessary to make $A/S$ a ring:

• For $A/S$ to be an additive group, $S$ must be an additive subgroup because $A$ is in particular an additive group.
• For $A/S$ to be a multiplicative semigroup, any $a\in A$ multiplied by $s\in S$ should go to zero in the quotient, hence $as\in S$.

Definition

Definition. An ideal $I\subset A$ is an additive subgroup such that $aI\in I$ for all $a\in A$.

Kinds

• Principal: ideal is generated by a single element
• Prime ideal
• Maximal ideal

Definition. Two ideals $a$ and $b$ are called coprime if $a+b=A$.

Theorem. ( Chinese Remainder Theorem) If $I_1,\ddots,I_k$ are pairwise coprime ideals, then

1. $I_1\cap\cdots\cap I_k=I_1\cdot \dots \cdot I_k$
2. $\phi:A\to\prod_i A/I_i$ is surjective

Common notations

Let $\mathcal{O}$ be an Integral domain, $k$ its field of fractions. Suppose $I,j$ are $\mathcal{O}$-submodules in $k$.

• $I^{-1}\coloneqq{x\in k\mid xI\subset\mathcal{O}}$
• $\mathcal{O}(I)\coloneqq{x\in k\mid xI\subset I}$
• $I+J\coloneqq{x+y\mid x\in I, y\in J}$
• $IJ\coloneqq{\sum x_iy_i\mid x_i\in I,y_i\in J}$ where the sum is finite.

Properties

1. There is a one-to-one order-preserving correspondence between the ideals $B$ of $A$ and the ideals $B'$ of $A/I$, given by $B=\phi^{-1}(B')$ where $\phi$ is the natural quotient $A\to A/I$.
2. The kernel and image of a ring homomorphism are ideals, the domain modulo the kernel is isomorphic to the image.