Mrs. Arkwright taught an adolescent Dave Lister, and had an affair with him, even having a "pregnancy scare" with Lister. This was sometime after Lister had already lost his virginity, aged twelve, with Michelle Fisher on Bootle Municipal Golf Course.
Mrs. Arkwright liked Lister's essay on glaciated valleys, and asked him to stay behind after class. Before Lister knew it, they were flattening the third-year's paper-mache Nativity scene. There was shepherds everywhere. She later believed that she may have been pregnant with Lister's child, but it turned out to be a false alarm.
Given that her name implied she was married, and that Rimmer sarcastically says that it was a "very formal relationship", it is possible that Mrs Arkwright had a husband.
Lister used Mrs Arkwright as the first example of a "pregnancy scare" Lister had, as they searching for the old letter from Hayley Summers that would prove whether Lister ever had become a father, or not. ("Dear Dave" deleted scenes, Series X)