Useful tricks:
- If \({\mathfrak{m}}\) is maximal and \(I\neq {\mathfrak{m}}\), then \(\left\langle{I, \mgm}\right\rangle = R\) is the entire ring.
Operations on ideals:
- \(I+J = \left\{{i+j{~\mathrel{\Big\vert}~}i\in I, j\in J}\right\}\) is the smallest ideal containing \(I\) and \(J\).
- \(\sum_k I_k = \left\{{\sum_k i_k {~\mathrel{\Big\vert}~}i_k\in I_k}\right\}\) where all but finitely many terms in each sum are zero.
- \(IJ = \left\langle{\left\{{ij}\right\}}\right\rangle = \left\{{\sum i_k j_k {~\mathrel{\Big\vert}~}i_k \in I, j_k\in J}\right\}\) is the ideal generated by all elementary product.
- \(I^n = \left\{{\prod_{k\leq n}x_k {~\mathrel{\Big\vert}~}x_k\in I}\right\}\).
- The distributive law \(a(b+c) = ab + ac\) for ideals does not generally hold. Instead there is a modular law: \begin{align*} a(b+c) = (a \cap b) + (a \cap c) \qquad \text{if } b \subseteq a \text{ or } c \subseteq a .\end{align*}
- In general, \(a \cap b = ab \iff a+b = \left\langle{1}\right\rangle\).
Show that if \(I, J{~\trianglelefteq~}{\mathbb{Z}}\) then \(I \cap J = IJ \iff\) their generators are coprime.
Show that TFAE:
- \(R\) is a field
- \(R\) is simple as an \(R{\hbox{-}}\)module
- Every ring morphism \(R\to S\) with \(S\) nonzero is injective and an isomorphism onto its image
- Show that maximal implies prime but not conversely.
- Show that inverse images of prime ideals are prime, but this fails for maximal ideals.
- Show that every non-unit is contained in a maximal ideal.
- Show that if \(I{~\trianglelefteq~}R\) and \(R\setminus I \subseteq R^{\times}\) then \(R\) is local and \(I\) is its maximal ideal.
- Let \(I{~\trianglelefteq~}R\coloneqq k[x_1, \cdots, x_{n}]\) and be the ideal \(I= xR\), the polynomials with zero constant term. Show that \(I\) is maximal, but is not principal for \(n>1\) and in fact requires at least \(n\) generators.
- Show that every nonzero prime in a PID is maximal.
- Show that \(\sqrt {0_R}\) is an ideal and \(R/\sqrt{0_R}\) is reduced.
- Show that \(\sqrt 0\) is the intersection of all prime ideals.
- Show that \(r\in {J ({R}) } \iff 1-rx\in R^{\times}\) for all \(x\in R\).
The category of algebras over a ring \(A\):
