Using the field of two elements $F_2$, let $S=F_2[M]$ where $M$ is the monoid of nonnegative real numbers under addition. The required ring is $R=S/I$ where $I$ is the ideal generated by elements of $M$ greater than $1$. (The elements look like linear combinations of elements from the interval $[0,1]$.)

Notes: $J(R)$ is idempotent and nil.

Keywords semigroup ring

- C. Hajarnavis and N. Norton. On dual rings and their modules. (1985) @ Example 6.2, pp 265-266
- N. C. Norton. Generalizations of the theory of quasi-frobenius rings. (1975) @ Example 3.2.2, p 112

Known Properties

Legend

- = has the property
- = does not have the property
- = information not in database

Name | Measure | |
---|---|---|

cardinality | $\mathfrak c$ | |

composition length | left: $\infty$ | right: $\infty$ |

Krull dimension (classical) | 0 |

Name | Description |
---|---|

Idempotents | $\{0,1\}$ |