Abstract Algebra Refresher: Review, Amplification, Examples

Two elements \(a,b\) in a ring (with identity) \(R\) are called associates if \(a = ubv\) for some units \(u,v\) in \(R\text{.}\) In a commutative ring, we can simply write \(a=bv.\)

Let \(R\) be a commutative ring with identity. Then two ideals \(I,J\) of \(R\) are said to be comaximal iff \(I+J = R.\)

Let \(X\) be a non-empty set. A relation on \(X\) is a subset \(R\subseteq X\times X\text{,}\) that is a collection of ordered pairs. Often instead of saying \((x,y)\in R\text{,}\) write \(x \sim y\) and say \(x\) is related to \(y.\)

An integral domain \(R\) is a Euclidean domain if it is equipped with a function (norm) \(d:R\setminus\{0\} \to \Z_{\ge 0}\) so that given two elements \(a,b \in R\) with \(b \ne0\text{,}\) there exist \(q,r \in R\) with \(a=bq+r\text{,}\) and either \(r=0\) or \(d(r) \lt d(b).\)

Let \(G\) be a group, and \(x\in G\text{.}\) Any positive integer \(n\) for which \(x^n = e\) is called an exponent for the element; the smallest exponent is called its order. The order may be infinite.

If there is a positive integer \(n\) so that \(x^n = e\) for every \(x \in G\text{,}\) then \(n\) is called an exponent for the group.

Let \(R\) be a ring. A subset \(I\subseteq R\) is called an ideal if \(I\) is an additive subgroup of \(R\text{,}\) with the property that \(R\cdot I \subseteq I\) and \(I\cdot R\subset I.\) This is also called a two-sided ideal. There are also left and right ideals in which only one condition holds. Of course in a commutative ring, all ideals are two-sided.

Recall that the property of being a (two-sided) ideal is precisely the condition required to make the quotient \(R/I\) a ring with well-defined operations on the cosets.

Let \(R\) be a ring. An element \(s\in R\) is an idempotent if \(s^2 = s.\)

A function \(f:X\to Y\) between sets \(X\) and \(Y\) is injective if for every \(x, x' \in X\text{,}\) \(f(x) = f(x')\) implies \(x=x'\text{.}\)

Let \(F\) denote the field of real or complex numbers. For \(z = a + bi \in \C\) (\(a,b \in \R\) and \(i^2 = -1\)), we have the notion of the complex conjugate of \(z\text{,}\) denoted \(\overline z = a-bi.\) Note that when \(z\in \R\text{,}\) that is \(z = a = a + 0i \in \C\text{,}\) we have \(z = \overline z.\) The magnitude (norm, \(absolute value\)) of \(z=a+bi\) is \(|z| = \sqrt{a^2 + b^2}.\)

An inner product space is a vector space \(V\) defined over a field \(F = \R\) or \(\C\) to which is associated an inner product. If \(F=\R\text{,}\) \(V\) is called a real inner product space, and if \(F=\C\text{,}\) then \(V\) is called a complex inner product space.

Let \(R\) be a commutative ring with identity. An element \(\pi \in R\) is said to be irreducible if \(\pi \ne 0\text{,}\) \(\pi \not\in R^\times\text{,}\) and whenever we write \(\pi = ab\text{,}\) either \(a\) or \(b\) is a unit in \(R.\)

Let \(R\) be a commutative ring with identity. An element \(\pi \in R\) is said to be prime if \(\pi \ne 0\text{,}\) \(\pi \not\in R^\times\text{,}\) and given \(a,b \in R\) with \(\pi \mid ab\text{,}\) then \(\pi \mid a\) or \(\pi \mid b.\)

We remark that in a noncommutative ring, a different definition is required: \(P\) is a prime ideal iff \(P\) is proper and for any ideals \(I,J \subset R\text{,}\) \(IJ \subseteq P\) implies \(I \subseteq P\) or \(J \subseteq P.\) If the ring is commutative, this definition is equivalent to the previous one.

Let \(R\) be a UFD, and \(p(x) = a_0 + a_1x+ \cdots + a_n x^n \in R[x].\) We say that \(p\) is a primitive polynomial if \(\gcd(a_1, \dots, a_n) =1,\) that is there is no common divisor of all the coefficients except for units. It is immediate that for any \(p \in R[x]\) that \(p = c(p) p_0\) where \(p_0\) is primitive and \(c(p) \in R\) which is usually referred to as the content of \(p\text{.}\)

It is a commutative ring if \(\times\) is commutative. The ring \(R\) has an identity if there is an element \(1\in R\) so that \(1\times r = r\times 1 = r\) for all \(r\in R.\) Generally, we write \(rs\) instead of \(r\times s.\)

For an element \(r \in R,\) \(-r\) is its additive inverse, and \(r^{-1}\) is its multiplicative inverse (if it exists). We denote by \(R^\times\) the unit group of a ring \(R\) with identity.

A nonempty subset \(H\) of a group \(G\) is a subgroup of \(G\) if it is closed under products and inverses. More succinctly, it is a subgroup if for every \(x,y \in H\text{,}\) \(xy^{-1} \in H.\) It is usually denoted \(H\le G\) or \(H \lt G\) for a proper subgroup.

A function \(f:X\to Y\) between sets \(X\) and \(Y\) is surjective if for every \(y\in Y\text{,}\) there exists an \(x \in X\) such that \(f(x)=y\text{.}\)

A nonzero element \(a\) in a ring \(R\) is a zero divisor if there is a nonzero \(b \in R\) with \(ab=0\) or \(ba=0\text{.}\) In noncommative rings, one condition may hold, but not the other (e.g., in matrix rings). If desired, one can talk about left and right zero divisors, though that will not be our focus.