Retrieving "Left Ideal" from the archives

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1. Quotient Ring

   Linked via "left ideal"

   $$(a+I)(b+I) = (ab) + I$$
   For these operations to be well-defined, it is crucial that $I$ be a two-sided ideal. If $I$ were merely a left ideal, the definition of multiplication would depend on the choice of representatives, leading to an ill-defined structure [1].
   The zero element of the quotient ring $R/I$ is the coset $0+I$, which is precisely the ideal $I$ itself. If $R$ has a multiplicative identity $1R$, the multiplicative identity of $R/I$ is $1R …