Annihilators of the Algebra of the Reduced Biquaternions

Introduction

This article reads the annihilators of the multiplication. For an element $\breve Q_{1}$ the annihilator is $$ \operatorname{Ann}(\breve Q_{1})=\{\breve Q_{2}:\breve Q_{1}\breve Q_{2}=0\}, $$ the set of elements the multiplication by $\breve Q_{1}$ kills. The result is a dichotomy: the annihilator of a unit is $\{0\}$, and the annihilator of a zero divisor is a whole idempotent line, $$ \operatorname{Ann}(Q_{+}\Pi_+)=\mathbb{C}\Pi_-,\qquad \operatorname{Ann}(Q_{-}\Pi_-)=\mathbb{C}\Pi_+\qquad(Q_{+},Q_{-}\neq0), $$ the other line, of real dimension two. The annihilator of $0$ is the whole algebra. Because the algebra is commutative, the left and the right annihilators coincide, and the annihilator of an element is a two-sided ideal; the two maximal ideals $\mathbb{C}\Pi_+$ and $\mathbb{C}\Pi_-$ are the annihilators of each other.

The article also reads the two-sided annihilator of the structure, the set of elements that annihilate the algebra, and records that it is $\{0\}$ because the algebra is unital. The two results are different: the annihilators of the elements are the two lines, and the annihilator of the algebra is trivial.

The results. The left and the right annihilators of an element coincide, since the multiplication is commutative, and they are an ideal. The annihilator of a unit is $\{0\}$; the annihilator of a nonzero element of one idempotent line is the other idempotent line, a complex line of real dimension two; the annihilator of $0$ is the whole algebra; and the annihilator of the whole algebra is $\{0\}$. The map $\breve Q\mapsto\operatorname{Ann}(\breve Q)$ takes the units and $0$ to the two trivial ideals and the two lines to each other, so the set of annihilators of the elements is the set of the four ideals of the algebra.

The Annihilator of an Element

Left and Right, They Coincide

The left and the right annihilators of an element are equal, because the multiplication is commutative.

Definition. For $\breve Q_{1}\in\mathbb{B}_{\mathrm{r}}$, $$ \operatorname{Ann}(\breve Q_{1})=\{\breve Q_{2}:\breve Q_{1}\breve Q_{2}=0\} $$ is the annihilator of $\breve Q_{1}$. Since the multiplication is commutative (Introduction to the Algebra of the Reduced Biquaternions), the left annihilator $\{\breve Q_{2}:\breve Q_{2}\breve Q_{1}=0\}$ and the right annihilator $\{\breve Q_{2}:\breve Q_{1}\breve Q_{2}=0\}$ are the same set, so the word needs no side.

Proposition. The annihilator of an element is a two-sided ideal of the algebra.

Proof. If $\breve Q_{1}\breve Q_{2}=0$ and $\breve Q_{1}\breve Q_{3}=0$ then $\breve Q_{1}(\breve Q_{2}+\breve Q_{3})=0$; if $\breve Q_{1}\breve Q_{2}=0$ and $\breve Q_{3}$ is any element then $\breve Q_{1}(\breve Q_{3}\breve Q_{2})=(\breve Q_{1}\breve Q_{3})\breve Q_{2}=(\breve Q_{3}\breve Q_{1})\breve Q_{2}=\breve Q_{3}(\breve Q_{1}\breve Q_{2})=0$, using the associativity and the commutativity. Hence $\operatorname{Ann}(\breve Q_{1})$ is closed under addition and under multiplication by an arbitrary element, that is an ideal.

The Annihilator of a Unit

The annihilator of a unit is $\{0\}$.

Theorem. If $\langle \breve Q_{1}\rangle_{\mathrm{ts}}\neq0$ then $\operatorname{Ann}(\breve Q_{1})=\{0\}$.

Proof. If $\breve Q_{1}\breve Q_{2}=0$ then multiplying by $\breve Q_{1}^{-1}$ gives $\breve Q_{2}=\breve Q_{1}^{-1}(\breve Q_{1}\breve Q_{2})=0$, and the inverse exists by Norm and Invertibility in the Algebra of the Reduced Biquaternions. So no nonzero element is annihilated by a unit.

The Annihilator of a Zero Divisor

The annihilator of a nonzero element of one idempotent line is the other idempotent line, a complex line of real dimension two.

Theorem (the annihilator of a line). For $Q_{+},Q_{-}\in\mathbb{C}^{\times}$, $$ \boxed{\; \operatorname{Ann}(Q_{+}\Pi_+)=\mathbb{C}\Pi_-,\qquad \operatorname{Ann}(Q_{-}\Pi_-)=\mathbb{C}\Pi_+ \;}. $$

Proof. Write $\breve Q_{2}=Q_{2\blacktriangle}+jQ_{2\blacktriangledown}$ and read it in the idempotent coordinates $\breve Q_{2}=Q_{2+}\Pi_++Q_{2-}\Pi_-$, with $Q_{2+}=Q_{2\blacktriangle}+Q_{2\blacktriangledown}$ and $Q_{2-}=Q_{2\blacktriangle}-Q_{2\blacktriangledown}$. Then $(Q_{+}\Pi_+)\breve Q_{2}=Q_{+}Q_{2+}\Pi_+$, because $\Pi_+^{2}=\Pi_+$ and $\Pi_+\Pi_-=0$ (Zero Divisors of the Algebra of the Reduced Biquaternions), and this is zero iff $Q_{2+}=0$, since $Q_{+}\neq0$ and a complex number has no zero divisors. The condition $Q_{2+}=0$ is $\breve Q_{2}=Q_{2\blacktriangle}-Q_{2\blacktriangle}j=Q_{2\blacktriangle}(1-j)=Q_{2-}\Pi_-$ with $Q_{2-}=2Q_{2\blacktriangle}\in\mathbb{C}$, that is $\breve Q_{2}\in\mathbb{C}\Pi_-$. The same computation on the other line.

Corollary (the two lines annihilate each other). The two maximal ideals are the annihilators of each other, $$ \boxed{\; \operatorname{Ann}(\mathbb{C}\Pi_+)=\mathbb{C}\Pi_-,\qquad \operatorname{Ann}(\mathbb{C}\Pi_-)=\mathbb{C}\Pi_+ \;}, $$ and each is a complex line of real dimension two.

Proof. The annihilator of a set is the intersection of the annihilators of its elements; every nonzero element of the line has the same annihilator, the other line, and the element $0$ of the line has the whole algebra, so the intersection is the other line.

The Annihilators of Zero and of the Subsets

The annihilator of $0$ is the whole algebra, and the annihilator of the set of all zero divisors is $\{0\}$.

Theorem. $\operatorname{Ann}(0)=\mathbb{B}_{\mathrm{r}}$, and $\operatorname{Ann}(\mathcal{N}\cup\{0\})=\{0\}$, where $\mathcal{N}$ is the null cone.

Proof. $0\cdot \breve Q_{2}=0$ for every $\breve Q_{2}$, so the annihilator of $0$ is the whole algebra. If $\breve Q_{2}$ annihilates every zero divisor, then in particular $\breve Q_{2}\Pi_+=0$ and $\breve Q_{2}\Pi_-=0$; expanding $\breve Q_{2}=Q_{2+}\Pi_++Q_{2-}\Pi_-$ gives $Q_{2+}=Q_{2-}=0$, so $\breve Q_{2}=0$.

The Two-Sided Annihilator of the Structure

The two-sided annihilator of the algebra is $\{0\}$, because the algebra is unital.

Definition. The two-sided annihilator of the structure is $$ \operatorname{Ann}(\mathbb{B}_{\mathrm{r}})=\{\breve Q_{2}:\breve Q_{1}\breve Q_{2}=0\ \text{for all }\breve Q_{1}\in\mathbb{B}_{\mathrm{r}}\}. $$

Theorem (the structure's annihilator is trivial). $\operatorname{Ann}(\mathbb{B}_{\mathrm{r}})=\{0\}$.

Proof. Take $\breve Q_{1}=1$, the unit: $1\cdot \breve Q_{2}=\breve Q_{2}$, so $\breve Q_{2}$ in the annihilator satisfies $\breve Q_{2}=0$. Conversely $0$ is annihilated by everything. The triviality is exactly the existence of a two-sided unit (Introduction to the Algebra of the Reduced Biquaternions), and it is the reason the interesting annihilators of this article are the per-element ones and not the structure's.

Remark (the two readings of the words). The phrase the two-sided annihilator of the structure has been read here as the annihilator of the ring; its value is $\{0\}$. The phrase the annihilator of the structure can also be read as the annihilator of the set of zero divisors, and that value is also $\{0\}$; and it can be read as the set of common annihilators of the two maximal ideals, which is $\mathbb{C}\Pi_+\cap\mathbb{C}\Pi_-=\{0\}$ as well. All three readings agree, and the single nonzero fact of the section is that the annihilators of the elements are the two lines while the annihilators of the structures are trivial.

The Lattice of Annihilators

The annihilators of the elements are the four ideals of the algebra, and the correspondence is the one of the table.

Theorem (the four ideals). The set $\{\operatorname{Ann}(\breve Q):\breve Q\in\mathbb{B}_{\mathrm{r}}\}$ is $\{0,\mathbb{C}\Pi_+,\mathbb{C}\Pi_-,\mathbb{B}_{\mathrm{r}}\}$, which is the set of all ideals of the algebra, and it is ordered by inclusion as the two chains $\{0\}\subset\mathbb{C}\Pi_+\subset\mathbb{B}_{\mathrm{r}}$ and $\{0\}\subset\mathbb{C}\Pi_-\subset\mathbb{B}_{\mathrm{r}}$, which share their two ends.

Proof. The table is read from the three theorems above. The ideals of the algebra are the four displayed (Zero Divisors of the Algebra of the Reduced Biquaternions, §The Maximal Ideals). The two lines are incomparable and both contain $\{0\}$ and are contained in $\mathbb{B}_{\mathrm{r}}$.

Corollary (the annihilator of the annihilator). $\operatorname{Ann}(\operatorname{Ann}(\breve Q))=\breve Q$ in the sense of ideals: the double annihilator of a unit is $\mathbb{B}_{\mathrm{r}}$, that of $Q_{+}\Pi_+$ is $\mathbb{C}\Pi_+$, that of $Q_{-}\Pi_-$ is $\mathbb{C}\Pi_-$, and that of $0$ is $\{0\}$.

Proof. $\operatorname{Ann}(\mathbb{B}_{\mathrm{r}})=\{0\}$ and $\operatorname{Ann}(\{0\})=\mathbb{B}_{\mathrm{r}}$ by the definition; and $\operatorname{Ann}(\mathbb{C}\Pi_-)=\mathbb{C}\Pi_+$ and $\operatorname{Ann}(\mathbb{C}\Pi_+)=\mathbb{C}\Pi_-$ are the corollary above. Reading the table twice reproduces the two lines.

Summary

The annihilator of an element is $\operatorname{Ann}(\breve Q_{1})=\{\breve Q_{2}:\breve Q_{1}\breve Q_{2}=0\}$; the multiplication is commutative, so the left and the right annihilators coincide and the annihilator of an element is a two-sided ideal. The annihilator of a unit is $\{0\}$; the annihilator of a nonzero element of one idempotent line is the other line, $$ \operatorname{Ann}(Q_{+}\Pi_+)=\mathbb{C}\Pi_-,\qquad \operatorname{Ann}(Q_{-}\Pi_-)=\mathbb{C}\Pi_+\qquad(Q_{+},Q_{-}\neq0), $$ a complex line of real dimension two; the annihilator of $0$ is the whole algebra; and the annihilator of the set of all zero divisors is $\{0\}$. The two maximal ideals $\mathbb{C}\Pi_+$ and $\mathbb{C}\Pi_-$ are the annihilators of each other.

The two-sided annihilator of the structure is trivial, $$ \operatorname{Ann}(\mathbb{B}_{\mathrm{r}})=\{\breve Q_{2}:\breve Q_{1}\breve Q_{2}=0\ \text{for all }\breve Q_{1}\}=\{0\}, $$ because the algebra is unital; and the three readings of the phrase — the annihilator of the ring, the annihilator of the set of zero divisors and the common annihilator of the maximal ideals — all give $\{0\}$. The set of annihilators of the elements is the set of the four ideals of the algebra, $\{0,\mathbb{C}\Pi_+,\mathbb{C}\Pi_-,\mathbb{B}_{\mathrm{r}}\}$, ordered by inclusion with the two lines incomparable; and the double annihilator returns the line, the algebra or the zero ideal according to the table.

Summary of Notation

Further Reading

• Zero Divisors of the Algebra of the Reduced Biquaternions, above this article in this category, for the two lines and the maximal ideals.
• Norm and Invertibility in the Algebra of the Reduced Biquaternions, above this article, for the criterion of invertibility and the group of units.
• Square Roots of One and the Idempotents of the Algebra of the Reduced Biquaternions, below this article, for the idempotents and the projectors.
• Ideals and Peirce Decomposition of the Algebra of the Reduced Biquaternions, below this article in this category, for the ideals the idempotents carry.
• Definitions for the Study of the Algebraic Structures over the Biquaternion $\mathbb{C}$ Space, for the definition of an ideal and of an annihilator.