The first female algebraist in US/Britain? Recently I dug up some biographical details of Lindsay Burch, of Hilbert-Burch Theorem fame, whose few papers have had quite an impact on commutative algebra. This made me curious about the first women who obtained PhDs in abstract algebra in the US and Britain.

Question 1: Who was the first woman to get a PhD in the US on a topic in algebra?

Potential answer: Mildred Sanderson, who obtained her PhD from Dickson in 1913 (published in Annals of Mathematics), and tragically died one year later. I found her name in Bell's "50 years of algebra in America, 1888-1938". Is that correct?
For Britain, I could not find any similar information. Burch got her PhD in 1967 under David Rees, but surely other women got British PhDs in algebra earlier. So:

Question 2: Who was the first woman to get a PhD in Great Britain on a topic in algebra?

 A: A few early 20th century US Ph.D.'s (the first three are before Mildred Sanderson, but after Annie MacKinnon):

*

*Ph.D. 1906 Bryn Mawr: Virginia Ragsdale, On the Arrangement of the Real Branches of Plane Algebraic
Curves


*Ph.D. 1909 Univ. Chicago: Anna
Johnson Pell Wheeler, Biorthogonal Systems of Functions with
Applications to the Theory of Integral Equations
It was Wheeler who brought Emmy Noether to Bryn Mawr in 1933, after her expulsion from Göttingen University. 


*Ph.D. 1911 UC Berkeley: Annie
Dale Biddle Andrews, Constructive theory of the unicursal plane
quartic by synthetic methods


*Ph.D. 1915 Univ. Chicago: Olive Hazlett,
On the Classification and Invariantive Characterization of Nilpotent
Algebras
Two late 19th century US Ph.D.'s who shared the advisor with Annie MacKinnon (although their research topic might not quite fit the algebra tag):

*

*Ph.D. 1893 Cornell: Ida Martha Metcalf, Geometric Duality in Space

*Ph.D. 1895 Cornell:
Agnes Sime Baxter, Abelian Integrals
A: I think a very honourable mention is the American Ida May Schottenfels, who did not receive a PhD, but who nevertheless was very active in mathematical research:

Ida May Schottenfels graduated from Northwestern University with a Bachelor of Arts degree in 1892. She then studied at Yale and the University of Chicago, earning a Masters degree in mathematics from Chicago in 1896. [...]  In 1910 she was appointed head of the department of mathematics in Toledo University.  In their study of "Women in the American Mathematical Research Community: 1891-1906" Fenster and Parshall cite Schottenfels as one of the two most "active" participants (along with Charlotte Angas Scott), listing her as giving 17 talks at mathematics conferences, publishing 3 papers, and attending 23 meetings and/or colloquia including those of the American Mathematical Society.

She was also the first mathematician to prove that there are two finite simple groups of the same order (namely of order $20160$). So although she did not receive a PhD, she surely seems to have deserved one!
A: I followed the reference suggested by KConrad in the comments and found perhaps the answer to Question 1:
Annie MacKinnon, who got her PhD from Cornell in 1894 with the thesis "Concomitant Binary Forms in Terms of the Roots" (invariant theory). The published version (Annals of Mathematics, 1894 - 1895, Vol. 9, No. 1/6 (1894 - 1895), pp. 95-157) can be found here.
As for Question 2, perhaps one can nominate Charlotte Scott as suggested in the comments, although from circumstantial evidence she probably did her thesis on geometry of plane curves, so I am still wondering if there is a more "algebraic" thesis later.
A: Christine Ladd-Franklin had her dissertation published but did not receive her PhD:

At Johns Hopkins Ladd developed her interest in symbolic logic through the lectures of Charles Sanders Peirce, writing a dissertation on "The Algebra of Logic" and publishing several more articles in The Analyst. However, Johns Hopkins did not allow women to receive the Ph.D. degree, so Ladd left the school in 1882 without that official recognition. Her dissertation, however, appeared in the volume Studies in Logic by Members of the Johns Hopkins University, edited by Charles S. Pierce, Little, Brown & Co., 1883 (pages 17-71).

(Not an answer, and too long for a comment, but still relevant to the question, in my humble opinion. I'm supposing, by the title of her dissertation, that it has something to do with algebra.)
