فرض کنید rیک حلقه ی جابجایی و یکدار باشد. تعمیم های متعددی از ایده آل های اول مورد بررسی قرار گرفته است. برای مثال یک ایده آل محض i از r به طور ضعیف اول است (به همین ترتیب، به طور تقریبی اول) اگر برای a,b متعلق به r که ab متعلق به i و مخالف صفراست، آنگاه a متعلق به i یا b متعلق به i. (به همین ترتیب اگر برای a,b متعلق به r که ab متعلق به i ولی متعلق به i بتوان دو نباشد، آنگاه a متعلق به i یا b متعلق به i). فرض کنید فی یک تابع باشد که از مجموعه ی تمام ایده آل های r به مجموعه ی تمام ایده آل های r به استثنای تهی تعریف شود. ایده آل محض i از r یک ایده آل فی - اول نامیده می شود اگر برای a,b متعلق به r که ab متعلق به r ولی متعلق به فی r نباشد، آنگاه a متعلق به r یا b متعلق به r باشد. نشان می دهیم که ایده آل های فی - اول دارای ویژگی های بسیار مشابهی با ایده آل های اول هستند.

the extent to which written corrective feedback on linguistic errors can play a role in helping l2 writers improve the accuracy of their writing continues to be an issue of interest to researchers and teachers since truscott (1996) mounted a case for its abolition. while there is growing empirical evidence that written corrective feedback can successfully target some types of linguistic error (e.g., bitchener & knoch, 2008; chandler, 2003; ferris, 2002; sheen, 2007), it is unclear which feedback strategy is more effective.the present study was an attempt to investigate the effect of both direct focused and recast written corrective feedback(wcf) on grammatical accuracy of efl learners writing achievement and retention. the target grammatical structure in this study was simple past verb (past tense copula verbs, irregular verbs and regular verbs, all of which were in the active voice). to this end ,60 intermediate female students from avid language school in tehran, were selected through preliminary english test (pet) and were randomly divided into two experimental groups ( direct focused and recast ). the study had a quasi-experimental design with pretests, immediate posttests and delayed posttests. three different picture compositions taken from byrne (1967) were used as narrative writing tests . one of the tests was administered in pre-test session , one in immediate post-test session and the other one in delayed post-test. for encouraging the participants to show their performance on the use of target grammatical structures, during treatment sessions,in both experimental groups, students completed three written narrative tasks separately. the findings showed that, the second experimental group (i.e., the recast group) outperformed the direct focused group. in addition, the lasting effect of recast wcf was more than the lasting effect of direct focused wcf on the grammatical accuracy of efl learners’ writing.

مطالعه حاضر بر آن است که نقش موفقیت و ماندگاری دو نوع بازخورد تصحیحی نوشتاری مستقیم متمرکز و ریکست را بر روی صحت دستوری نوشتار زبان آموزان ایرانی بررسی کند