Ideals

Let \(R\) be a ring. An ideal \(S\) of \(R\) is a subset \(S \subset R\) such that:
a) \(S\) is closed under addition: If \(a, b \in S\), then \(a + b \in S\)
b) The zero element of \(R\) is in \(S\): \(0 \in S\)
c) \(S\) is closed under additive inverses: If \(a \in S\), then \(-a\in S\)
d) If \(r \in R\) and \(x \in S\), then \(r x \in S\) and \(x r \in S\). In other words, \(S\) is closed under multiplication (on either side) by arbitrary ring elements.

Subring vs Ideal

The difference between a subring and an ideal is that a subring must be closed under multiplication of elements in the subring. On the other hand, an ideal must be closed under multiplication of an element in the ideal by any element in the ring.
Since the ideal definition requires more multiplicative closure than the subring definition, every ideal is a subring, but the converse is not true.
Ideals are used to construct quotient rings, similar to how normal subgroups are used to construct quotient groups.

Left and Right Ideals

Left Ideal: In a noncommutative ring \(R\), a left ideal is a subset \(I\) which is an additive subgroup of \(R\) and such that for all \(r \in R\) and all \(a \in I\),

\[ra \in I\]

Right Ideal: In a noncommutative ring \(R\), a right ideal is a subset \(I\) which is an additive subgroup of \(R\) and such that for all \(r \in R\) and all \(a \in I\),

\[ar \in I\]

In a commutative ring, the right and left ideals coincide.

Proper and Nontrivial Ideals

Lemma: Let \(R\) be a ring. Then \(R\) and \(\{0\}\) are ideals.

A proper ideal is an ideal other than \(R\).
A nontrivial ideal is an ideal other than \(\{0\}\).

Principal Ideals

Let \(R\) be a commutative ring, and let \(a \in R\). The principal ideal generated by \(a\) is:

\[\langle a \rangle = \{r a \mid r \in R\}\]

For example, in the ring of polynomials with real coefficients \(R[x]\), this is the principal ideal generated by \(x^2 + 4\):

\[\langle x^2 + 4 \rangle = \{(x^2 + 4)\cdot f(x) \mid f(x) \in R[x]\}\]

It is the set consisting of all multiples of \(x^2 + 4\). For example, here are some elements of \(\langle x^2 + 4 \rangle\): \((2 x + 5)\cdot (x^2 + 4), \quad (-\pi x^{50} + \sqrt{2})\cdot (x^2 + 4), \quad 0 = 0\cdot (x^2 + 4)\)

Lemma: Let \(R\) be a commutative ring, and let \(a \in R\). Then \(\langle a \rangle\) is a two-sided ideal in \(R\).

Prime and Maximal Ideals

Prime Ideal

An ideal \(I\) in a commutative ring \(R\) is said to be a prime ideal if, for arbitrary element \(r\), \(s\) of \(R\):
\(r \cdot s \in J\) implies either \(r \in J\) or \(s \in J\).

Maximal Ideal

A proper ideal \(I\) in a commutative ring \(R\) is called maximal if there exists no proper ideal in \(R\) which properly contains \(I\).

Quotient Rings

If \(R\) is a ring and \(I\) is a two-sided ideal, the quotient ring of \(R\) mod \(I\) is the group of cosets \(\dfrac{R}{I}\) with the operations of coset addition and coset multiplication.

Let \(R\) be a ring, and let \(I\) be an ideal:
a) If \(R\) is a commutative ring, so is \(R/I\)
b) If \(R\) has a multiplicative identity 1, then \(1 + I\) is a multiplicative identity for \(R/I\). In this case, if \(r \in R\) is a unit, then so is \(r + I\), and \((r +I)^{-1} = r^{-1} + I\).

Euclidean Rings

A Euclidean Ring is a commutative ring \(R\) having the property that to each \(x \in R\), a non-negative integer \(\theta (x)\) can be assigned such that:

i) \(\theta (x) = 0\) if and only if \(x=z\), the zero element of \(R\)
ii) \(\theta (x \cdot y) \geq \theta(x)\) when \(x \cdot y \neq z\)
iii) For every \(x \in R\) and \(y \neq z \in R\),
\(x = y \cdot q + r\), \(q, r \in R\), \(0 \leq \theta(r) \leq \theta(y)\)

PYQs

Ideals

1) Let \(R^{C}=\) ring of all real value continuous functions on \([0,1]\) under the operations \((f+g) x=f(x)+g(x)\), \((f g) x=f(x) g(x)\). Let \(M=\left\{f \in R^{C} / f\left(\dfrac{1}{2}\right)=0\right\}\). Is \(M\) a maximal ideal of \(R\)? Justify your Answer.

[2013, 15M]

2) Is the ideal generated by 2 and \(X\) in the polynomial ring \(Z[X]\) of polynomials in a single variable \(X\) with coefficients in the ring of integers \(Z\), a principal ideal? Justify your answer.

[2012, 15M]

3) Describe the maximal ideals in the ring of Gaussian integers \(Z[i]=\left\{a+ bi \vert a, b \in Z\right\}\).

[2012, 20M]

4) How many proper, non-zero ideals does the ring \(Z_{12}\) have? Justify your answer. How many ideals does the ring \(Z_{12} \oplus Z_{12}\) have? Why?

[2009, 2+3+4+6=15M]

5) Let \(R=\left[ \begin{array}{cc}{a} & {b} \\ {c} & {d}\end{array}\right]\) where \(a, b, c, d \in Z\). Show that \(R\) is a ring under matrix addition and multiplication \(\left\{A=\left[ \begin{array}{ll}{a} & {0} \\ {b} & {0}\end{array}\right], a, b \in Z\right\}\). Then show that \(A\) is a left ideal of \(R\) but not a right ideal of \(R\).

[2007, 12M]

6) Let \(M\) and \(N\) be two ideals of a ring \(R\). Show that \(M \cup N\) is an ideal of \(R\) if and only if either \(M \subseteq N\) or \(N \subseteq M\).

[2003, 10M]

7) Let \(R\) be the ring of all real-valued continuous functions on the closed interval \([0,1]\). Let \(M=\left\{f(x) \in R / f\left(\dfrac{1}{3}\right)=0\right\}\). Show that \(M\) is a maximal ideal of \(R\).

[2003, 10M]

8) Prove that in the ring \(F[x]\) of polynomial over a field \(F\), the ideal \(1= \vert p(x) \vert\) is maximal if and only if the polynomial \(p(x)\) is irreducible over \(F\).

[2002, 20M]

9) If \(R\) is a commutative ring with unit element and \(M\) is an ideal of \(R\), then show that \(M\) is maximal ideal of \(R\) if and only if \(R/M\) is a field.

[2001, 20M]

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