Principal ideal

 ( Ideals ) 

In mathematics, specifically ring theory, a principal ideal is an ideal $I$ in a ring $R$ that is generated by a single element $a$ of $R$ through multiplication by every element of $R.$ The term also has another, similar meaning in order theory, where it refers to an (order) ideal in a poset $P$ generated by a single element $x\; \backslash in\; P,$ which is to say the set of all elements less than or equal to $x$ in $P.$

The remainder of this article addresses the ring-theoretic concept.

---

Definitions

• A left principal ideal of $R$ is a subset of $R$ given by $Ra\; =\; \backslash \{ra\; :\; r\; \backslash in\; R\backslash \}$ for some element $a.$
• A right principal ideal of $R$ is a subset of $R$ given by $aR\; =\; \backslash \{ar\; :\; r\; \backslash in\; R\backslash \}$ for some element $a.$
• A two-sided principal ideal of $R$ is a subset of $R$ given by $RaR\; =\; \backslash \{r\_1\; a\; s\_1\; +\; \backslash ldots\; +\; r\_n\; a\; s\_n:\; r\_1,s\_1,\; \backslash ldots,\; r\_n,\; s\_n\; \backslash in\; R\backslash \}$ for some element $a,$ namely, the set of all finite sums of elements of the form $ras.$

While the definition for two-sided principal ideal may seem more complicated than for the one-sided principal ideals, it is necessary to ensure that the ideal remains closed under addition.

If $R$ is a commutative ring, then the above three notions are all the same. In that case, it is common to write the ideal generated by $a$ as $\backslash langle\; a\; \backslash rangle$ or $(a).$

---

Examples and non-examples

• The principal ideals in the (commutative) ring $\backslash mathbb\{Z\}$ are $\backslash langle\; n\; \backslash rangle\; =\; n\backslash mathbb\{Z\}=\backslash \{\backslash ldots,-2n,-n,0,n,2n,\backslash ldots\backslash \}.$ In fact, every ideal of $\backslash mathbb\{Z\}$ is principal (see ).
• In any ring $R$, the sets $\backslash \{0\backslash \}=\; \backslash langle\; 0\backslash rangle$ and $R=\backslash langle\; 1\backslash rangle$ are principal ideals.
• For any ring $R$ and element $a,$ the ideals $Ra,aR,$ and $RaR$ are respectively left, right, and two-sided principal ideals, by definition. For example, $\backslash langle\; \backslash sqrt\{-3\}\; \backslash rangle$ is a principal ideal of $\backslash mathbb\{Z\}\backslash sqrt\{-3\}.$
• In the commutative ring $\backslash mathbb\{C\}x,y$ of complex number polynomials in two variables, the set of polynomials that vanish everywhere on the set of points $\backslash \{(x,y)\backslash in\backslash mathbb\{C\}^2\backslash mid\; x=0\backslash \}$ is a principal ideal because it can be written as $\backslash langle\; x\backslash rangle$ (the set of polynomials divisible by $x$).
• In the same ring $\backslash mathbb\{C\}x,y$, the ideal $\backslash langle\; x,\; y\backslash rangle$ generated by both $x$ and $y$ is not principal. (The ideal $\backslash langle\; x,\; y\backslash rangle$ is the set of all polynomials with zero for the constant term.) To see this, suppose there existed a generator $p$ for $\backslash langle\; x,y\backslash rangle,$ so $\backslash langle\; x,\; y\backslash rangle=\backslash langle\; p\backslash rangle.$ Then $\backslash langle\; p\backslash rangle$ contains both $x$ and $y,$ so $p$ must divide both $x$ and $y.$ Then $p$ must be a nonzero constant polynomial. This is a contradiction since $p\backslash in\backslash langle\; p\backslash rangle$, but the only constant polynomial in $\backslash langle\; x,\; y\backslash rangle$ is the zero polynomial.
• In the ring $\backslash mathbb\{Z\}\backslash sqrt\{-3\}\; =\; \backslash \{a\; +\; b\backslash sqrt\{-3\}:\; a,\; b\backslash in\; \backslash mathbb\{Z\}\; \backslash \},$ the numbers where $a\; +\; b$ is even form a non-principal ideal. This ideal forms a regular hexagonal lattice in the complex plane. Consider $(a,b)\; =\; (2,0)$ and $(1,1).$ These numbers are elements of this ideal with the same norm (two), but because the only units in the ring are $1$ and $-1,$ they are not associates.

---

Related definitions A ring in which every ideal is principal is called principal, or a principal ideal ring. A principal ideal domain (PID) is an integral domain in which every ideal is principal. Any PID is a unique factorization domain; the normal proof of unique factorization in the (the so-called fundamental theorem of arithmetic) holds in any PID.

As an example, $\backslash mathbb\{Z\}$ is a principal ideal domain, which can be shown as follows. Suppose $I=\backslash langle\; n\_1,\; n\_2,\; \backslash ldots\backslash rangle$ where $n\_1\backslash neq\; 0,$ and consider the surjective homomorphisms $\backslash mathbb\{Z\}/\backslash langle\; n\_1\backslash rangle\; \backslash rightarrow\; \backslash mathbb\{Z\}/\backslash langle\; n\_1,\; n\_2\backslash rangle\; \backslash rightarrow\; \backslash mathbb\{Z\}/\backslash langle\; n\_1,\; n\_2,\; n\_3\backslash rangle\backslash rightarrow\; \backslash cdots.$ Since $\backslash mathbb\{Z\}/\backslash langle\; n\_1\backslash rangle$ is finite, for sufficiently large $k$ we have $\backslash mathbb\{Z\}/\backslash langle\; n\_1,\; n\_2,\; \backslash ldots,\; n\_k\backslash rangle\; =\; \backslash mathbb\{Z\}/\backslash langle\; n\_1,\; n\_2,\; \backslash ldots,\; n\_\{k+1\}\backslash rangle\; =\; \backslash cdots.$ Thus $I=\backslash langle\; n\_1,\; n\_2,\; \backslash ldots,\; n\_k\backslash rangle,$ which implies $I$ is always finitely generated. Since the ideal $\backslash langle\; a,b\backslash rangle$ generated by any integers $a$ and $b$ is exactly $\backslash langle\; \backslash mathop\{\backslash mathrm\{gcd\}\}(a,b)\backslash rangle,$ by induction on the number of generators it follows that $I$ is principal.

---

Properties Any Euclidean domain is a PID; the algorithm used to calculate greatest common divisors may be used to find a generator of any ideal. More generally, any two principal ideals in a commutative ring have a greatest common divisor in the sense of ideal multiplication. In principal ideal domains, this allows us to calculate greatest common divisors of elements of the ring, up to multiplication by a unit; we define $\backslash gcd(a,\; b)$ to be any generator of the ideal $\backslash langle\; a,\; b\; \backslash rangle.$

For a Dedekind domain $R,$ we may also ask, given a non-principal ideal $I$ of $R,$ whether there is some extension $S$ of $R$ such that the ideal of $S$ generated by $I$ is principal (said more loosely, $I$ becomes principal in $S$). This question arose in connection with the study of rings of algebraic integers (which are examples of Dedekind domains) in number theory, and led to the development of class field theory by Teiji Takagi, Emil Artin, David Hilbert, and many others.

The principal ideal theorem of class field theory states that every integer ring $R$ (i.e. the ring of integers of some number field) is contained in a larger integer ring $S$ which has the property that every ideal of $R$ becomes a principal ideal of $S.$ In this theorem we may take $S$ to be the ring of integers of the Hilbert class field of $R$; that is, the maximal unramified abelian extension (that is, Galois extension whose Galois group is abelian group) of the fraction field of $R,$ and this is uniquely determined by $R.$

Krull's principal ideal theorem states that if $R$ is a Noetherian ring and $I$ is a principal, proper ideal of $R,$ then $I$ has height at most one.

---

See also

• Ascending chain condition for principal ideals

• Joseph A. Gallian (2025). 9781305657960, Cengage Learning. ISBN 9781305657960

Post Comment