{"product_id":"rings-close-to-regular-9789048161164","title":"Rings Close to Regular","description":"\u003cp\u003e • Author(s): A. A. Tuganbaev\u003cbr\u003e • Publisher: Springer\u003cbr\u003e • Publisher Imprint: Springer\u003cbr\u003e • BISAC: Algebra - Abstract\u003c\/p\u003e\u003cp\u003e\u003c\/p\u003e\u003cp\u003ePreface All rings are assumed to be associative and (except for nilrings and some stipulated cases) to have nonzero identity elements. A ring A is said to be regular if for every element a E A, there exists an element b E A with a = aba. Regular rings are well studied. For example, [163] and [350] are devoted to regular rings. A ring A is said to be tr-regular if for every element a E A, there is an element n b E A such that an = anba for some positive integer n. A ring A is said to be strongly tr-regular if for every a E A, there is a positive integer n with n 1 n an E a + An Aa +1. It is proved in [128] that A is a strongly tr-regular ring if and only if for every element a E A, there is a positive integer m with m 1 am E a + A. Every strongly tr-regular ring is tr-regular [38]. If F is a division ring and M is a right vector F-space with infinite basis {ei} l' then End(MF) is a regular (and tr-regular) ring that is not strongly tr-regular. The factor ring of the ring of integers with respect to the ideal generated by the integer 4 is a strongly tr-regular ring that is not regular.\u003c\/p\u003e","brand":"Springer","offers":[{"title":"Paperback","offer_id":45283209576599,"sku":"9789048161164","price":3639.0,"currency_code":"INR","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0666\/3471\/1191\/files\/9789048161164.webp?v=1769304374","url":"https:\/\/atlanticbooks.com\/products\/rings-close-to-regular-9789048161164","provider":"Atlantic Books","version":"1.0","type":"link"}