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I admit, I haven't gotten really far in studying abstract algebra, but I was always curious (ever since I saw a definition of an ideal) why is the notion of left-sided ideal introduced when we quotient by two-sided ideals? Are there examples of use of exclusively left (left but not two-sided) ideals in ring theory?

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When the concept of a left $R$-module is introduced (an analogue to a vector space over a field), it turns out that the ring $R$ has a natural structure of left $R$-module (the same way a field is one-dimensional vector space over itself). Then the left ideals of $R$ are precisely its submodules and the quotient has a structure of an $R$-module as well. – Pavel Čoupek Apr 17 '14 at 19:08
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Right ideals are submodules of the module right $R_R$, but two-sided ideals are bisubmodules of $_RR_R$. They are completely different in flavor.

The distinction is pointless in commutative algebra, of course, but in general noncommutative algebra it is important. For example, $M_n(\Bbb R)$ has infinitely many right ideals but only two two-sided ideals.

Another example is the ring of linear transformations of a vector space over a field which has countable dimension. Not only does the ring have infinitely many right ideals, but it doesn't even satisfy the ascending chain condition on right ideals. On the other hand, it has exactly three two-sided ideals.

So as you can see, one-sided and two-sided ideals can display very different behaviors and have very different implications about their ring.

Using an ideal $I$, the cosets $R/I$ form a ring. However if $I$ were only a one-sided ideal and not two sided, you could not do the same thing. The best you can do is to give the quotient a module structure, in that case.

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A good example for studying left-ideals instead of two-sided ideals are matrix rings, which are not commutative (in general), so that there is a difference. For an extensive discussion about left ideals and two-sided ideals in matrix rings, see What are the left and right ideals of matrix ring? How about the two sided ideals?

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