Commutative Rings, Fields and Ordered Fields: Standard Notation

¶ This setting extends The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion: its conventions and notation are in force.

¶ For a commutative ring $R$, with $+$, $\cdot$, $0$ and $1$, and $x,y\in R$: $-x$ and $x-y$ are as in Negatives, Differences, Reciprocals and Quotients §negative; for $n\in\mathbb{N}_{0}$, $x^{n}$ is as in Powers with Exponents in the Natural Numbers with Zero §power and Powers with Exponents in the Natural Numbers with Zero §zero; and finite sums and products of elements of $R$ are as in Sums and Products over a Finite Set and over an Interval §operation, Sums and Products over a Finite Set and over an Interval §empty, Sums and Products over a Finite Set and over an Interval §subsets, Sums and Products over a Finite Set and over an Interval §intervals and Sums and Products over a Finite Set and over an Interval §sums, which applies because $+$ and $\cdot$ are associative and commutative with neutral elements $0$ and $1$ by Commutative Rings §ring, on both sides by commutativity. By A Binary Operation Has at Most One Neutral Element §unique they are the only neutral elements of $+$ and $\cdot$, so the $1$ of Powers with Exponents in the Natural Numbers with Zero §zero is the unit of $R$, $x^{0}=1$, and the empty sum and the empty product of Sums and Products over a Finite Set and over an Interval §empty are $0$ and $1$.

¶ As permitted by The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion §overloading, an element $n$ of $\mathbb{N}_{0}$, introduced as such or given by a numeral, standing where an element of a commutative ring $R$ is required, in particular as an operand next to an element of $R$ or as a side of an equation or inequality whose other side lies in $R$, denotes its image $n_{R}$ in $R$. Positions that require an element of $\mathbb{N}_{0}$, such as exponents, subscripts and the limits of sums and products, are excepted; there, $n-m$ for $m\le n$ is the difference in $\mathbb{N}_{0}$. An equation or inequality between elements of $\mathbb{N}_{0}$ alone is read in $\mathbb{N}_{0}$, unless it is said to hold in $R$, in which case each element of $\mathbb{N}_{0}$ in it, other than in the positions excepted above, denotes its image in $R$. The reading is unambiguous: $0_{R}$ and $1_{R}$ are the zero and the unit of $R$ by The Image of the Natural Numbers with Zero in a Commutative Ring Respects Zero, One, Sums, Products, Differences, Powers, and Finite Sums and Products §constants; the image of a sum $m+n$ or a product $mn$ formed in $\mathbb{N}_{0}$ is the sum or product in $R$ of the images, by The Image of the Natural Numbers with Zero in a Commutative Ring Respects Zero, One, Sums, Products, Differences, Powers, and Finite Sums and Products §sum and The Image of the Natural Numbers with Zero in a Commutative Ring Respects Zero, One, Sums, Products, Differences, Powers, and Finite Sums and Products §product; the image of a difference $n-m$ with $m\le n$ or of a power $m^{k}$ formed in $\mathbb{N}_{0}$ is $n_{R}-m_{R}$, respectively $(m_{R})^{k}$, by The Image of the Natural Numbers with Zero in a Commutative Ring Respects Zero, One, Sums, Products, Differences, Powers, and Finite Sums and Products §difference and The Image of the Natural Numbers with Zero in a Commutative Ring Respects Zero, One, Sums, Products, Differences, Powers, and Finite Sums and Products §power; and the image of a finite sum or product formed in $\mathbb{N}_{0}$, such as $\prod_{k=1}^{n}k$, is the corresponding sum or product in $R$ of the images, by The Image of the Natural Numbers with Zero in a Commutative Ring Respects Zero, One, Sums, Products, Differences, Powers, and Finite Sums and Products §finite.

¶ If $R$ is a field, then for $x,y\in R$ with $y\neq0$, $y^{-1}$ and $x/y$ are as in Negatives, Differences, Reciprocals and Quotients §reciprocal.

¶ If $R$, with a total order $\le$, is an ordered ring, then the order notation of Sets and Maps: Ordinary Notation §orders is used for $\le$, and positive, nonnegative, negative and nonpositive elements are as in Positive, Nonnegative, Negative and Nonpositive Elements of an Ordered Ring §sign, and the rules of arithmetic and order of Rules of Order in an Ordered Ring: Differences, Sums, Negatives, Products and Squares are in force.

¶ If $R$, with a total order $\le$, is an ordered field, then $|x|$ is as in Absolute Value in an Ordered Field §absolute-value, and the rules of arithmetic and order of Rules of Arithmetic and Order in an Ordered Field are in force.