NIELIT 2016 DEC Scientist B (IT) - Section B: 21

Question

Consider the set $S=\{1,\omega,\omega ^2​\}$, where $\omega$ and $\omega^2​$ are cube roots of unity. If $*$ denotes the multiplication operation, the structure $(S,*)$ forms:

• A. A group
• B. A ring
• C. An integral domain
• D. A field

Answer 1

$*$	$1$	$w$	$w^{2}$
$1$	$1$	$w$	$w^{2}$
$w$	$w$	$w^{2}$	$1$
$w^{2}$	$w^{2}$	$1$	$w$

A Group is an algebraic structure which satisfies
1) closure
2) Associativity
3) Have Identity element
4) Invertible
Over '*' operation the $ S$ = {$1$, $w$, $w^{2}$ } satisfies the above properties.
The identity element is $1$ and inverse of $1$ is$ 1$, inverse of$ w$ is $w^{2}$ and inverse of$ w^{2}$ is $w$

https://gateoverflow.in/1150/gate2010-4

https://www.geeksforgeeks.org/gate-gate-cs-2010-question-4/

Comments

Does it follow commutative property?

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What is the meaning of ring here ?