{'fraction_non_alphanumeric': 0.04624664879356569, 'fraction_numerical': 0.012734584450402145, 'mean_word_length': 3.665625, '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}}
Weakly primitive graded rings

The strongest generalization of Jacobson's classical density theorem for prime rings is the theorem of Zelmanowitz [1] giving a complete description of weakly primitive rings. We shall give a description in the graded case based on the graded version of Zelmanowitz' theorem in [2]. Let G be a semigroup, H a group, and A an (H, G)-polygon. Following [3], the polygon A G is said to be strictly exact if, whenever ag 1 = ag 2 (a E A, g 1 ,g 2 E G), it follows that g 1 = g 2 . We shall assume that A is a strictly exact right G-polygon and a strictly exact left H-polygon, and that the action of G on it is injective, that is, if a 1 g = a 2 g, then a 1 = a 2 . In what follows, R is a G-graded ring, M is an A-graded right R-module, END(M R ) is an H-graded ring of endomorphisms, and h(R) is the set of homogeneous elements of R. Rings graded by a group G are studied in the monograph [4], [5] deals with rings and modules graded by a semigroup, and modules graded by a set on which a group G acts are considered in [6]. We recall some definitions. A module M is said to be critically compressible if it can be embedded by a homogeneous homomorphism in each of its non-zero graded submodules but not in any of its proper graded factor-modules. A graded ring is said to be weakly primitive if it has a faithful critically compressible graded module. A graded ring R is said to be prime if, for graded ideals I and J, it follows from IJ = 0 that I = 0 or J = 0.