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Let 풰 be an algebra over the field 픽. We say that 풰 is two-sided zero product determined if every bilinear functional ϕ : 풰×풰→ 픽 the following holds: if ϕ(x,y) = 0 whenever xy = yx = 0, then there exist linear functionals F1 and F2 on 풰 such that ϕ(x,y) = F1(xy)+F2(yx) for all x,y ∈풰. We show that the unital triangular algebra 풯 = 풜ℳ0 ℬ is a two-sided zero product determined algebra if and only if 풜 and ℬ are two-sided zero product determined algebras, and then we get various results about this property for generalized triangular algebras and block upper triangular matrix algebras. We also provide an application of the main result to determine the structure of commutativity preserving maps at commutative zero products on triangular algebras. We note that some of the previous results about the two-sided zero product determined property have been generalized. [ABSTRACT FROM AUTHOR]

Background/Objectives: The pathogenesis of preadolescent acne has not been well studied, and it is uncertain if Cutibacterium acnes is a predominant organism in the microbiome in this age group. The aim of this study was to analyze the microbiome of preadolescent females and to assess whether benzoyl peroxide impacts the microbiome. Methods: The study enrolled girls, aged 7‐12 years, with evidence of at least six acne lesions who had not been previously treated. Participants' skin surface of forehead, cheeks, nose, chin, left retroauricular crease, and extruded contents of a comedonal lesion were sampled at baseline. Participants used benzoyl peroxide 4% wash for 6‐8 weeks and returned for skin surface sampling and extraction collection. Microbiome analysis was performed using 16S ribosomal RNA gene amplicon sequencing on all swab and lesional extraction samples. Results: Fifty‐one participants were enrolled with a median IGA score of 2 (mild). Changes in microbiome diversity were associated with increasing age and number of acne lesions (P = 0.001). C. acnes had higher abundances on forehead and nose, as opposed to cheeks and chin (P = 0.009). Bacterial diversity (alpha diversity) of the skin microbiome was comparable between preadolescent at baseline and after treatment with benzoyl peroxide. Conclusion: This is the first large assessment characterizing female acne microbiome in early and late preadolescence. Results show that preadolescent acne can vary in its microbial profile, reflecting surrounding changes associated with the onset of puberty. Although benzoyl peroxide use was associated with decreased acne counts, its effect on microbial diversity was not demonstrated in our study. [ABSTRACT FROM AUTHOR]