One-field Landau–Ginzburg Jacobian algebra

Let $\mathbb{C}$ be the complex field and let $\mathbb{C}[X]$ be the polynomial ring of finitely supported complex coefficient sequences, with coefficientwise addition and convolution multiplication. The field structure makes $\mathbb{C}$ a commutative ring, and these operations make $\mathbb{C}[X]$ a commutative ring; this ring structure is the one used in the quotient below. Let $W\in \mathbb{C}[X]$. Define the formal derivative by linearity and $(X^n{)}^{\mathrm{\prime }}=n{X}^{n-1}$ for $n\ge 1$, with $1^{\mathrm{\prime }}=0$. The notation $(W^{\mathrm{\prime }}(X))$ means the ideal generated by the single polynomial $W^{\mathrm{\prime }}(X)$; in this commutative polynomial ring it is exactly $\{Q(X)W^{\mathrm{\prime }}(X):Q\in \mathbb{C}[X]\}$. The one-field Jacobian algebra is the quotient ring

$$\mathrm{Jac}(W):=\mathbb{C}[X]\mathrm{/}(W^{\mathrm{\prime }}(X))\mathrm{.}$$

This is a purely algebraic definition. Its use as the physical protected chiral ring of a Landau–Ginzburg theory requires the separate chiral-ring prescription stated in the model item; the quotient definition alone makes no such identification.

The complex numbers as $\mathbb{R}[x]\mathrm{/}(x^2+1)$, with the real embedding and imaginary unit $i$

$\mathbb{C}=\mathbb{R}[x]\mathrm{/}(x^2+1)$ is a field, every element is uniquely $a+bi$, and every nonzero element has inverse $(a-bi)\mathrm{/}(a^2+b^2)$

Every field is a commutative ring with $1\mathrm{\ne }0$; it is an integral domain, and it is a commutative division ring

The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution

Polynomial convolution makes $R[x]$ a commutative ring containing $R$ as its constant subring

The ideal generated by a subset and principal ideals

In a commutative ring, $(S)$ consists of finite sums $\sum r_i{s}_i$, and $(a)=Ra$

The quotient ring $R\mathrm{/}I$ with $(r+I)(s+I)=rs+I$