Abstract Algebra
《Abstract Algebra》 · 第 224—225 页 · Chapter 7，Section 7.1，Example (5)
(The (real) Hamilton Quaternions) Let $\mathbb H$ be the collection of elements of the form $a+bi+cj+dk$ where $a,b,c,d\in\mathbb R$ are real numbers (loosely, "polynomials in $1,i,j,k$ with real coefficients") where addition is defined "componentwise" by $(a+bi+cj+dk)+(a'+b'i+c'j+d'k)=(a+a')+(b+b')i+(c+c')j+(d+d')k$ and multiplication is defined by expanding $(a+bi+cj+dk)(a'+b'i+c'j+d'k)$ using the distributive law (being careful about the order of terms) and simplifying using the relations $i^2=j^2=k^2=-1$, $ij=-ji=k$, $jk=-kj=i$, $ki=-ik=j$ (where the real number coefficients commute with $i,j$ and $k$). […] The Hamilton Quaternions are a noncommutative ring with identity $(1=1+0i+0j+0k)$.