Ring Ideals: Structure, Quotients, Operations, and Extensions

All flashcards

Two-sided ideal

An additive subgroup closed under multiplication by ring elements on both the left and the right.

Equivalence modulo an ideal

For an ideal $I$, elements $x$ and $y$ are equivalent when $x-y\in I$.

Kernel of a ring homomorphism

The set of elements mapped to zero; it is a two-sided ideal.

Primary ideal in a commutative ring

An ideal $I$ such that $ab\in I$ implies $a\in I$ or $b^n\in I$ for some positive integer $n$.

Fractional ideal

An $R$-submodule of the quotient field that multiplication by some nonzero element of $R$ sends into $R$.

Quiz questions

1. For a ring $R$ and an ideal $I$, what condition ensures multiplication of cosets makes $R/I$ a ring?Answer: $I$ is a two-sided ideal of $R.
2. If $f:R\to S$ is a ring homomorphism, what kind of ideal is $\ker(f)$?Answer: A two-sided ideal of $R.
3. In a commutative ring, what is the quotient by a maximal ideal?Answer: A field.
4. If $I+J=R$ for ideals in a commutative ring, how are their product and intersection related?Answer: $IJ=I\cap J$.
5. If $M$ is a finitely generated module and $J(R)M=M$, what does Nakayama's lemma imply?Answer: $M=0$.