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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:

1. A group
2. A ring
3. An integral domain
4. A field

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

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/

What is the meaning of ring here ?

1 vote