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Let $R$ be a ring. Let $I\subseteq R$ such that

  1. $I+I\subseteq I$

  2. $IR\subseteq I$

  3. $RI\subseteq I$.

Such an $I$ is called ideal of $R$.

Can you check that is the definition true?

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    How can a definition be true or not? It's just a definition. Or are you asking whether it coincides with classical definition? Then yes, it does. As in with the definition of "two-sided" ideal. – 2017-02-21
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    @freakish Thanks. I did not understand second and third property that are these same? – 2017-02-21
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    They are not the same. Not in non-commutative rings. – 2017-02-21
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    @freakish Okey thanks. – 2017-02-21
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    I could, now how about you do? – 2017-02-21

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If you add $I\ne\emptyset$, it coincides with the usual definition of two-sided ideal as it is given for a non-commutative ring with unity $R$. However, in non-commutative ring theory, the notions of left ideal and right ideal get quite some usage.