We will cover following topics

Rings

A ring is an abelian group $R$ with binary operation $+$ (“addition”), together with a second binary operation $\cdot$ (“multiplication”). The operations satisfy the following axioms:

i) Multiplication is associative: For all $a, b, c \in R$ ,

(a \cdot b) \cdot c = a \cdot (b \cdot c)

ii) The Distributive Law holds: For all $a, b, c \in R$ ,

a \cdot (b + c) = a \cdot b + a \cdot c and (a + b) \cdot c = a \cdot c + b \cdot c

In other words, if we expand the definition of abelian group, then we can say that a non-empty set $R$ is said to form a ring with respect to the binary operations addition $(+)$ and multiplication $(\cdot)$ provided, for arbitrary $a$ , $b$ , $c$ $\in R$ , the following properties hold:

Associative Law of addition

$(a + b) + c = a + (b + c)$
Commutative Law of addition

$a + b = b + a$
Existence of an additive identity (zero)

There exists $z \in R$ such that $a + z = a$
Existence of additive inverses

For each $a \in R$ , there exists $- a \in R$ such that $a + (- a) = z$
Associative Law of multiplication

$(a \cdot b) \cdot c = a \cdot (b \cdot c)$
Distributive Laws

$a (b + c) = a \cdot b + a \cdot c$
$(b + c) a = b \cdot a + c \cdot a$

$(b + c) a = b \cdot a + c \cdot a$

The systems $Z$ , $R$ , $Q$ and $C$ are some examples of rings.

Types of Rings

Ring with Identity

A ring $R$ has a multiplicative identity if there is an element $1 \in R$ such that $1 \neq 0$ , and such that for all $a \in R$ ,

1 \cdot a = a and a \cdot 1 = a

A ring satisfying this axiom is called a ring with 1, or a ring with identity.

Note that in the term “ring with identity”, the word “identity” refers to a multiplicative identity. Every ring has an additive identity (“0”) by definitin.

Commutative Ring

A ring for which multiplication is commutative is called a commutative ring.

Thus, if R is a ring and $a b = b a$ for all $a, b \in R$ , then $R$ is a commutative ring.

Note that the adjective “commutative” applies to the multiplication operation; the addition operation is always commutative by definition.

Polynomial Ring

A polynomial ring is a ring which is formed from the set of polynomials in one or more indeterminates (also called variables) with coefficients in another ring.

Ring with Unity

A ring having a multiplicative identity element (unit element or unity) is called a ring with identity element or ring with unity.

Product Ring

Let $R$ and $S$ be rings. The product ring $R \times S$ of $R$ and $S$ is the set consisting of all ordered pairs $(r, s)$ , where $r \in R$ and $s \in S$ . Addition and multiplication are defined component-wise: For $a, b \in R$ and $x$ , $y \in S$ ,

(a, x) + (b, y) = (a + b, x + y)

(a, x) \cdot (b, y) = (a \cdot b, x \cdot y)

The additive identity is $(0, 0)$ ; the additive inverse $- (r, s)$ of $(r, s)$ is $(- r, - s)$ .

Subrings

Let $R$ be a ring. A non-empty subset $S$ of the set $R$ , which is itself a ring with respect to the binary operations on $R$ , is called a subring of $R$ .

Theorem

Let $R$ be a ring and $S$ be a proper subset of the set $R$ .

Then $S$ is a subring of $R$ if and only if:

i) $S$ is closed with respect to the ring operations

ii) for each $a \in S$ , we have $- a \in S$

Homomorphism of Rings

Let $R$ and $S$ be rings. A ring homomorphism is a function $f : R \to S$ such that:

a) For all $x, y \in R$ , $f (x + y) = f (x) + f (y)$
b) For all $x, y \in R$ , $f (x y) = f (x) f (y)$

Usually, we require that if $R$ and $S$ are rings with 1, then

c) $f (1_{R}) = 1_{S}$

The first two properties stipulate that $f$ should “preserve” the ring structure - addition and multiplication.

Lemma 1: Let $R$ and $S$ be rings and let $f : R \to S$ be a ring homomorphism.

a) $f (0) = 0$ b) $f (- r) = - f (r)$ for all $r \in R$

Lemma 2: Let $R$ , $S$ , and $T$ be rings, and let $f : R \to S$ and $g : S \to T$ be ring homomorphisms. Then the composite $g \cdot f : R \to T$ is a ring homomorphism.

Kernel and Image

Kernel: The kernel of a ring map $ϕ : R \to S$ is

k e r ϕ = {r \in R ∣ ϕ (r) = 0}

Image: The image of a ring map $\phi: R \rightarrow S$ is

i m ϕ = {ϕ (r) ∣ r \in R}

The kernel of a ring map is like the null space of a linear transformation of vector spaces. The image of a ring map is like the column space of a linear transformation.
The kernel of a ring map is a two-sided ideal and image of ring map is a subring of $S$ .

Isomorphism of Rings

Let $R$ and $S$ be rings. A ring isomorphism from $R$ to $S$ is a bijective ring homomorphism $f : R \to S$ .
If there is a ring isomorphism $f : R \to S$ , $R$ and $S$ are isomorphic and this relationship is symbolically represented as $R \approx S$ .

For example, if $R$ is a ring, the identity map $id : R \to R$ is an isomorphism of $R$ with itself.

Since a ring isomorphism is a bijection, isomorphic rings must have the same cardinality. So, for example, $Z_{6} ≉ Z_{42}$ , because the two rings have different numbers of elements.

However, $Z$ and $Q$ have the “same number” of elements (the same cardinality), but they are not isomorphic as rings. (Intuition: $Z$ is a field, while $Q$ is only an integral domain.)

PYQs

Rings

1) Give an example of a ring having identity but a subring of this having a different identity.

[2015, 10M]

2) Prove that the set $Q (\sqrt{5}) = {a + b \sqrt{5} : a, b \in Q}$ is commutative ring with identity.

[2014, 15M]

3) Let $F$ be the set of all real valued continuous functions defined on the closed interval $[0, 1]$ . Prove that $(F, +, .)$ is a Commutative Ring with unity with respect to addition and multiplication of functions defined point wise as below:-
$(f + g) x = f (x) + g (x)$ and $(f g) x = f (x) g (x)$
$x \in [0, 1]$ where $f, g \in F$ .

[2011, 15M]

4) Show that the quotient ring $\frac{Z [i]}{1 + 3 i}$ is isomorphic to the ring $\frac{Z}{10 Z}$ where $Z [i]$ denotes the ring of Gaussian integers.

[2010, 15M]

5) Suppose that there is a positive even integer $n$ such that $a^{n} = a$ for all the elements $a$ of some ring $R$ . Show that $a + a = 0$ for all $a \in R$ and $a + b = 0 \Rightarrow a = b$ for all $a, b \in R$ .

[2008, 12M]

6) Show that in the ring $R = {a + b \sqrt{- 5} | a, b \in Z}$ , the elements $α = 3$ and $β = 1 + 2 \sqrt{- 5}$ are relatively prime, but $α γ$ and $β γ$ have no g.c.d in $R$ , where $γ = 7 (1 + 2 \sqrt{- 5})$ .

[2007, 15M]

7) Show that $(1 + i)$ is a prime element in the ring $R$ of Gaussian integers.

[2005, 12M]

8) Prove that if the cancellation law holds for a ring $R$ then $a (\neq 0) \in R$ is not a zero divisor and conversely.

[2004, 10M]

< Previous Next >