中文摘要：

结合广义逆理论研究了环中平等投影（EP）元、正规元和对称元的性质和一些等价刻画.给出了在核逆存在的情况下元素为EP元的一些等价条件.设a∈R,那么a是EP元当且仅当aa a^#=a^#aa .同时,讨论了正则元是EP元的等价刻画.设a∈R,那么存在b∈R,使得a=a ba且a是EP元当且仅当a∈R,a=aba.同样地,给出了在核逆存在的情况下元素为正规元的一些等价条件.设a∈R,那么a是正规元当且仅当a^＊a =a a^＊.而且在群逆和Moore-Penrose逆存在的情况下给出了元素为正规元和对称元的一些涉及次数的等价条件.设a∈R ＋∩R^#,且存在n∈N,那么a是正规元当且仅当a^＊a^＋（a^#）^n=a^#a^＊（a^＋）^n.结果推广了Mosi等人的结论.

英文摘要：

The properties and some equivalent characterizations of equal projection（ EP）, normal and Hermitian elements in a ring are studied by the generalized inverse theory. Some equivalent conditions that an element is EP under the existence of core inverses are proposed. Let a∈R , then a is EP if and only if aa a^# = a^#aa . At the same time, the equivalent characterizations of a regular element to be EP are discussed.Let a∈R, then there exist b∈R such that a = aba and a is EP if and only if a∈R , a = a ba. Similarly, some equivalent conditions that an element is normal under the existence of core inverses are proposed. Let a∈R , then a is normal if and only if a^＊a = a a^＊. Also, some equivalent conditions of normal and Hermitian elements in rings with involution involving powers of their group and Moore-Penrose inverses are presented. Let a∈R ∩R^#, n∈N, then a is normal if and only if a^＊ a^＋（ a^#） n = a^# a＊（ a^＋） ^n. The results generalize the conclusions of Mosiet al.