Ideals: Detailed Explanation, Proofs, and Derivations
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. Ideals generalize certain subsets in the ring of integers such as even numbers, which form an ideal in the ring of all integers.
Definitions
Let \( R \) be a ring. A subset \( I \subseteq R \) is called a left ideal if:
1. \( I \) is an additive subgroup of \( R \),
2. For any \( r \in R \) and \( x \in I \), the product \( rx \in I \).
Similarly, \( I \) is called a right ideal if:
1. \( I \) is an additive subgroup of \( R \),
2. For any \( r \in R \) and \( x \in I \), the product \( xr \in I \).
A subset \( I \subseteq R \) is called a two-sided ideal (or simply an ideal) if it is both a left and a right ideal.
Examples
1. The set of even integers \( 2\mathbb{Z} \) is an ideal in the ring of integers \( \mathbb{Z} \).
2. The set of all polynomials with zero constant term forms an ideal in the ring of all polynomials.
Principal Ideals
An ideal \( I \) is called a principal ideal if there exists an element \( a \in R \) such that:
$$ I = (a) = \{ ra \mid r \in R \} $$
Properties of Ideals
1. The intersection of any collection of ideals of \( R \) is also an ideal.
2. The sum of two ideals \( I \) and \( J \) is defined as:
$$ I + J = \{ x + y \mid x \in I, y \in J \} $$
This sum is also an ideal.
3. The product of two ideals \( I \) and \( J \) is defined as:
$$ IJ = \left\{ \sum_{k=1}^n x_k y_k \mid x_k \in I, y_k \in J, n \in \mathbb{N} \right\} $$
This product is also an ideal.
Quotient Rings
Given a ring \( R \) and an ideal \( I \subseteq R \), the quotient ring \( R/I \) is the set of cosets of \( I \) in \( R \), with addition and multiplication defined as:
$$ (r + I) + (s + I) = (r + s) + I $$
$$ (r + I)(s + I) = rs + I $$
The quotient ring \( R/I \) inherits many properties from \( R \).
Prime Ideals and Maximal Ideals
An ideal \( P \subseteq R \) is called a prime ideal if \( P \neq R \) and whenever \( ab \in P \) for \( a, b \in R \), then either \( a \in P \) or \( b \in P \).
An ideal \( M \subseteq R \) is called a maximal ideal if \( M \neq R \) and there is no ideal \( J \) such that \( M \subset J \subset R \).
Proofs and Derivations
Proof that the Intersection of Ideals is an Ideal
Let \( \{I_\alpha\}_{\alpha \in A} \) be a collection of ideals in a ring \( R \). Define \( I = \cap_{\alpha \in A} I_\alpha \). We need to show that \( I \) is an ideal.
1. Since each \( I_\alpha \) is an additive subgroup of \( R \), the intersection \( I \) is also an additive subgroup of \( R \).
2. Let \( r \in R \) and \( x \in I \). Then \( x \in I_\alpha \) for all \( \alpha \in A \). Since each \( I_\alpha \) is a left ideal, \( rx \in I_\alpha \) for all \( \alpha \in A \). Thus, \( rx \in I \). Therefore, \( I \) is a left ideal.
Similarly, we can show that \( I \) is a right ideal. Hence, \( I \) is an ideal.
Proof that the Sum of Ideals is an Ideal
Let \( I \) and \( J \) be ideals in a ring \( R \). Define \( K = I + J \). We need to show that \( K \) is an ideal.
1. Since \( I \) and \( J \) are additive subgroups of \( R \), their sum \( K \) is also an additive subgroup of \( R \).
2. Let \( r \in R \) and \( x \in K \). Then \( x = i + j \) for some \( i \in I \) and \( j \in J \). Since \( I \) and \( J \) are left ideals, \( ri \in I \) and \( rj \in J \). Therefore, \( r(i + j) = ri + rj \in K \). Hence, \( K \) is a left ideal.
Similarly, we can show that \( K \) is a right ideal. Therefore, \( K \) is an ideal.
Proof that the Product of Ideals is an Ideal
Let \( I \) and \( J \) be ideals in a ring \( R \). Define \( K = IJ \). We need to show that \( K \) is an ideal.
1. Since \( I \) and \( J \) are additive subgroups of \( R \), their product \( K \) is also an additive subgroup of \( R \).
2. Let \( r \in R \) and \( x \in K \). Then \( x = \sum_{k=1}^n x_k y_k \) for some \( x_k \in I \) and \( y_k \in J \). Since \( I \) and \( J \) are left ideals, \( rx_k \in I \) and \( y_k \in J \). Therefore, \( r \sum_{k=1}^n x_k y_k = \sum_{k=1}^n (rx_k)y_k \in K \). Hence, \( K \) is a left ideal.
Similarly, we can show that \( K \) is a right ideal. Therefore, \( K \) is an ideal.
