{'fraction_non_alphanumeric': 0.037889039242219216, 'fraction_numerical': 0.018944519621109608, 'mean_word_length': 3.9006622516556293, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Algebras Generated by Two Quadratic Elements

Let K be a field of any characteristic and let R be an algebra generated by two elements satisfying quadratic equations. Then R is a homomorphic image of F = K ⟨x, y | x 2 + ax + b = 0, y 2 + cy + d = 0⟩ for suitable a, b, c, d ∈ K. We establish that F can be embedded into the 2 × 2 matrix algebra with entries from the polynomial algebra over the algebraic closure of K and that F and satisfy the same polynomial identities as K-algebras. When the quadratic equations have double zeros, our result is a partial case of more general results by Ufnarovskij, Borisenko, and Belov from the 1980s. When each of the equations has different zeros, we improve a result of Weiss, also from the 1980s.