There was no indication in the record that Alva received a divorce. Clearly in 1880 Mary resided in Henry Co, OH as head of house with her children. There is undoubtedly more to this story, but here's what I have found so far.
14 February 2011
Divorce Action in Williams County, Ohio - Rice (Roll 48)
Alva Rice v Mary Read Rice (Journal 15 p444; Roll 48 Box 166 case number 2148) - 7 Dec 1889
By Pamela Pattison Lash
The couple was married in Perry Co, OH c1854. Alva could not give an accurate description of dates because of the accidental destruction of the family Bible. He said the couple had the following children:
Anna, age 14
Martha, age 18, married
Lizzie, age 20, married
Mary J, age 22, married
Arthur, age 23
Peter, age 27
William, age 29
Sarah, age 30, married (Daniel Haley in NW Twp, Williams Co, OH)
Alva charged Mary with gross neglect and being willfully absent for more than four years. He resided in Hamler, Henry Co, OH and wanted custody of the minor, Anna.
Alva, the son of William and Mary Tharp Rice, was born in Ohio on 25 Sept/Oct 1829. He died in Carryall Twp, Paulding Co, OH on 4 Oct 1906.
Mary Read was born on 21 Jan 1833 in Ohio.
1860 Perry, Hocking, Ohio; Roll: M653_988; Page: 203; Image: 410; Family History Library Film: 803988.
h/h 36/36 Rice, Alba 40 OH cooper
Mary 26 OH
Sarah Catharine 9 OH
William 7 OH
Peter 1 OH
1870 Harrison, Henry, Ohio; Roll: M593_1221; Page: 295A; Image: 108; Family History Library Film: 552720.
h/h 126/127 Rice, Alvin 40 OH farmer
Polly 40 OH
William 16 OH
Peter 13 OH
Arthur 9 OH
Mary J 7 OH
Lizzie 3 OH
Alba Rice paid chattel tax in Harrison Twp, Henry Co, OH c1872
1880 Marion, Henry, Ohio; Roll: 1032; Family History Film: 1255032; Page: 196D; Enumeration District: 119; Image: 0395.
h/h 28/28 Rice, Mary 47 OH-PA-PA
Arthur son 19 OH-OH-OH works in stave factory
Jane dau 16 OH-OH-OH
Lizzie dau 13 OH-OH-OH
Martha dau 10 OH-OH-OH
Stella dau 7 OH-OH-OH (5 June 1873, Harrison Twp, Henry Co, OH Birth – Zedelia Rice; died Jan 1883)
Anna dau 4 OH-OH-OH
1880 Defiance, Defiance, Ohio; Roll: 1011; Family History Film: 1255011; Page: 104C; Enumeration District: 233; Image: 0467.
h/h 401/462 Rice, Alva 50 boarder with John and Maria Lehman family OH-VA-PA can not read or write, widower
1900 Marion, Henry, Ohio; Roll: T623_1286; Page: 6A; Enumeration District: 29.
h/h 117/117 Rice, Mary (Jan 1833) 67 mar 48Y 10/6 ch OH-PA-PA owns farm
1900 Antwerp, Paulding, Ohio; Roll: T623_1312; Page: 7B; Enumeration District: 46.
h/h 181/184 Rice, Alva brother-in-law of George Long and wife M. Ellen (Oct 1829) 70 widower, shoemaker, OH-OH-OH