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Why in the following problem the author take the left ideal in the following form? could anyone explain this for me please?
2026-03-25 19:05:02.1774465502
Why in the following problem the author take the left ideal in the following form?
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The important thing to understand is that because the module is cyclic, it means there exist some element, $a$ here, that generates the entire module when THE $R$-action is applied to it. After that it is simply that he wants to identify two elements if their difference action on $a$ gives the identity.
Assume for the arguement that $r$ and $r'$ both in $R$ and no $0$ and $x\in R/J$ generates $R/J$. Then if we have $rx=r'x=0$ it follows that $(r-r')x=0$, as none of those are the zero element, we get that $r-r'\in J$. So that is how we know we must form the ideal in the form he did.