Unfortunately, I cannot confirm the relationship between Ann Ryan Carrigan and Mary Ryan Nolan. Ann Ryan was the wife of Patrick Carrigan and they had 9 children (see 1880 census). Two of these children were Michael and Mary.
Ann was the daughter of John Ryan, who was born in County Tipperary, Ireland.
Ann also possibly had a sister, Margaret, who married Francis Doubet in Peoria County on May 16, 1842.
Ann died on Feb 11, 1901 in Kickapoo. I have found her in the the following census years: 1850; 1860; 1870; 1880; 1891; 1900.