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In this paper, we characterize the idempotency, cleanness, and unit-regularity of the commutator [E1, E2] = E1E2 - E2E1 involving idempotents E1, E2 in certain subrings of M2(Z).

In this paper, we study the permutation properties of the class of trinomials of the form f (x) = x4q+1 + λ1xq+4 + λ2x2q+3 ∈ Fq2 [x] , where λ1, λ2 ∈ Fq and they are not simultaneously zero. We find all necessary and suﬀicient conditions on λ1 and λ2 such that f (x) permutes Fq2 , where q is odd and q = 22k+1, k ∈

Mayenite (Ca12Al14O33) has remarkable properties such as high oxygen mobility, ionic conductivity, and catalytic activity. It has many different applications, including oxide-conducting electrolytes, fluorescent lamps, moisture sensors, hydrogen-permeable membranes, oxygen pumps, hydrogen storage, and catalysis. However, pure and homogeneous mayenite synthesis parameters have not yet been fully explored. This study examines the effect of synthesis parameters including metal salt (MS) to citric acid (CA) molar ratios (1:1 and 1:2), pH (0.4-2), and calcination temperature (900-1200 degrees C) in citrate sol-gel method on the crystal structure of mayenite. Synthesized materials were examined by thermogravimetric (TG), X-ray diffraction (XRD), X-ray photoelectron spectroscopy (XPS), Raman spectroscopy, N2 adsorption-desorption, scanning electron microscopy (SEM), inductively coupled plasma-optical emission spectrometry (ICP-OES), Fourier-transform infrared spectroscopy (FTIR), and pyridine adsorbed diffuse reflectance Fourier-transformed infrared spectroscopy (DRIFTS) analyses. The results show that all 3 parameters contribute to the mayenite phase formation and different impurity phases can be observed depending on the synthesis parameters. With no pH adjustment and an MS to CA ratio of 1, other phases of calcium aluminate mostly form. Mayenite becomes the main phase by doubling the CA amount. Besides CA, pH is also an important factor in mayenite synthesis. When the pH was adjusted to 2 with the MS to CA ratio at 1:1, mayenite was formed as the main phase, but other phases of calcium aluminate were also observed in the structure. XRD results show that all parameters studied influence the crystal structure of the final material, including the calcination temperature. This study shows that pure mayenite can be synthesized with a calcination temperature of 1200 degrees C, at a pH of 2, and the MS to CA molar ratio of 1:2.

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