## 10.3 Basic notions

The following is a list of basic notions in commutative algebra. Some of these notions are discussed in more detail in the text that follows and some are defined in the list, but others are considered basic and will not be defined. If you are not familiar with most of the italicized concepts, then we suggest looking at an introductory text on algebra before continuing.

$R$ is a

*ring*,$x\in R$ is

*nilpotent*,$x\in R$ is a

*zerodivisor*,$x\in R$ is a

*unit*,$e \in R$ is an

*idempotent*,an idempotent $e \in R$ is called

*trivial*if $e = 1$ or $e = 0$,$\varphi : R_1 \to R_2$ is a

*ring homomorphism*,$\varphi : R_1 \to R_2$ is

*of finite presentation*, or*$R_2$ is a finitely presented $R_1$-algebra*, see Definition 10.6.1,$\varphi : R_1 \to R_2$ is

*of finite type*, or*$R_2$ is a finite type $R_1$-algebra*, see Definition 10.6.1,$\varphi : R_1 \to R_2$ is

*finite*, or*$R_2$ is a finite $R_1$-algebra*,$R$ is a

*(integral) domain*,$R$ is

*reduced*,$R$ is

*Noetherian*,$R$ is a

*principal ideal domain*or a*PID*,$R$ is a

*Euclidean domain*,$R$ is a

*unique factorization domain*or a*UFD*,$R$ is a

*discrete valuation ring*or a*dvr*,$K$ is a

*field*,$L/K$ is a

*field extension*,$L/K$ is an

*algebraic field extension*,$\{ t_ i\} _{i\in I}$ is a

*transcendence basis*for $L$ over $K$,the

*transcendence degree*$\text{trdeg}(L/K)$ of $L$ over $K$,the field $k$ is

*algebraically closed*,if $L/K$ is algebraic, and $\Omega /K$ an extension with $\Omega $ algebraically closed, then there exists a ring map $L \to \Omega $ extending the map on $K$,

$I \subset R$ is an

*ideal*,$I \subset R$ is

*radical*,if $I$ is an ideal then we have its

*radical*$\sqrt{I}$,$I \subset R$ is

*nilpotent*means that $I^ n = 0$ for some $n \in \mathbf{N}$,$I \subset R$ is

*locally nilpotent*means that every element of $I$ is nilpotent,$\mathfrak p \subset R$ is a

*prime ideal*,if $\mathfrak p \subset R$ is prime and if $I, J \subset R$ are ideal, and if $IJ\subset \mathfrak p$, then $I \subset \mathfrak p$ or $J \subset \mathfrak p$.

$\mathfrak m \subset R$ is a

*maximal ideal*,any nonzero ring has a maximal ideal,

the

*Jacobson radical*of $R$ is $\text{rad}(R) = \bigcap _{\mathfrak m \subset R} \mathfrak m$ the intersection of all the maximal ideals of $R$,the ideal $(T)$

*generated*by a subset $T \subset R$,the

*quotient ring*$R/I$,an ideal $I$ in the ring $R$ is prime if and only if $R/I$ is a domain,

an ideal $I$ in the ring $R$ is maximal if and only if the ring $R/I$ is a field,

if $\varphi : R_1 \to R_2$ is a ring homomorphism, and if $I \subset R_2$ is an ideal, then $\varphi ^{-1}(I)$ is an ideal of $R_1$,

if $\varphi : R_1 \to R_2$ is a ring homomorphism, and if $I \subset R_1$ is an ideal, then $\varphi (I) \cdot R_2$ (sometimes denoted $I \cdot R_2$, or $IR_2$) is the ideal of $R_2$ generated by $\varphi (I)$,

if $\varphi : R_1 \to R_2$ is a ring homomorphism, and if $\mathfrak p \subset R_2$ is a prime ideal, then $\varphi ^{-1}(\mathfrak p)$ is a prime ideal of $R_1$,

$M$ is an

*$R$-module*,for $m \in M$ the

*annihilator*$I = \{ f \in R \mid fm = 0\} $ of $m$ in $R$,$N \subset M$ is an

*$R$-submodule*,$M$ is an

*Noetherian $R$-module*,$M$ is a

*finite $R$-module*,$M$ is a

*finitely generated $R$-module*,$M$ is a

*finitely presented $R$-module*,$M$ is a

*free $R$-module*,if $0 \to K \to L \to M \to 0$ is a short exact sequence of $R$-modules and $K$, $M$ are free, then $L$ is free,

if $N \subset M \subset L$ are $R$-modules, then $L/M = (L/N)/(M/N)$,

$S$ is a

*multiplicative subset of $R$*,the

*localization*$R \to S^{-1}R$ of $R$,if $R$ is a ring and $S$ is a multiplicative subset of $R$ then $S^{-1}R$ is the zero ring if and only if $S$ contains $0$,

if $R$ is a ring and if the multiplicative subset $S$ consists completely of nonzerodivisors, then $R \to S^{-1}R$ is injective,

if $\varphi : R_1 \to R_2$ is a ring homomorphism, and $S$ is a multiplicative subsets of $R_1$, then $\varphi (S)$ is a multiplicative subset of $R_2$,

if $S$, $S'$ are multiplicative subsets of $R$, and if $SS'$ denotes the set of products $SS' = \{ r \in R \mid \exists s\in S, \exists s' \in S', r = ss'\} $ then $SS'$ is a multiplicative subset of $R$,

if $S$, $S'$ are multiplicative subsets of $R$, and if $\overline{S}$ denotes the image of $S$ in $(S')^{-1}R$, then $(SS')^{-1}R = \overline{S}^{-1}((S')^{-1}R)$,

the

*localization*$S^{-1}M$ of the $R$-module $M$,the functor $M \mapsto S^{-1}M$ preserves injective maps, surjective maps, and exactness,

if $S$, $S'$ are multiplicative subsets of $R$, and if $M$ is an $R$-module, then $(SS')^{-1}M = S^{-1}((S')^{-1}M)$,

if $R$ is a ring, $I$ an ideal of $R$, and $S$ a multiplicative subset of $R$, then $S^{-1}I$ is an ideal of $S^{-1}R$, and we have $S^{-1}R/S^{-1}I = \overline{S}^{-1}(R/I)$, where $\overline{S}$ is the image of $S$ in $R/I$,

if $R$ is a ring, and $S$ a multiplicative subset of $R$, then any ideal $I'$ of $S^{-1}R$ is of the form $S^{-1}I$, where one can take $I$ to be the inverse image of $I'$ in $R$,

if $R$ is a ring, $M$ an $R$-module, and $S$ a multiplicative subset of $R$, then any submodule $N'$ of $S^{-1}M$ is of the form $S^{-1}N$ for some submodule $N \subset M$, where one can take $N$ to be the inverse image of $N'$ in $M$,

if $S = \{ 1, f, f^2, \ldots \} $ then $R_ f = S^{-1}R$ and $M_ f = S^{-1}M$,

if $S = R \setminus \mathfrak p = \{ x\in R \mid x\not\in \mathfrak p\} $ for some prime ideal $\mathfrak p$, then it is customary to denote $R_{\mathfrak p} = S^{-1}R$ and $M_{\mathfrak p} = S^{-1}M$,

a

*local ring*is a ring with exactly one maximal ideal,a

*semi-local ring*is a ring with finitely many maximal ideals,if $\mathfrak p$ is a prime in $R$, then $R_{\mathfrak p}$ is a local ring with maximal ideal $\mathfrak p R_{\mathfrak p}$,

the

*residue field*, denoted $\kappa (\mathfrak p)$, of the prime $\mathfrak p$ in the ring $R$ is the field of fractions of the domain $R/\mathfrak p$; it is equal to $R_\mathfrak p/\mathfrak pR_\mathfrak p = (R \setminus \mathfrak p)^{-1}R/\mathfrak p$,given $R$ and $M_1$, $M_2$ the

*tensor product*$M_1 \otimes _ R M_2$,given matrices $A$ and $B$ in a ring $R$ of sizes $m \times n$ and $n \times m$ we have $\det (AB) = \sum \det (A_ S)\det ({}_ SB)$ in $R$ where the sum is over subsets $S \subset \{ 1, \ldots , n\} $ of size $m$ and $A_ S$ is the $m \times m$ submatrix of $A$ with columns corresponding to $S$ and ${}_ SB$ is the $m \times m$ submatrix of $B$ with rows corresponding to $S$,

etc.

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