Rings in which each finitely generated right ideal is automorphism-invariant (right$fa$-rings) are shown to be isomorphic to a formal matrix ring. Among other results it is also shown that (i) if $R$ is a right nonsingular ring and $n>1$ is an integer, then $R$ is a right self injective regular ring if and only if the matrix ring ${𝕄}_n(R)$ is a right $fa$-ring, if and only if ${𝕄}_n(R)$ is a right automorphism-invariant ring and (ii) a right nonsingular ring $R$ is a right $fa$-ring if and only if $R$ is a direct sum of a square-full von Neumann regular right self-injective ring and a strongly regular ring containing all invertible elements of its right maximal ring of fractions. In particular, we show that a right semiartinian (or left semiartinian) ring $R$ is a right nonsingular right $fa$-ring if and only if $R$ is a left nonsingular left $fa$-ring.

每个有限生成的右理想环都是自同构不变的（右$F\mathrm{一个}$-rings ) 被证明与形式矩阵环同构。除其他结果外，还表明（i）如果$R$是一个右非奇异环并且$n>1$是一个整数，那么$R$是右自内射正则环当且仅当矩阵环${𝕄}_n(R)$是一种权利$F\mathrm{一个}$-环，当且仅当${𝕄}_n(R)$是一个右自同构不变环和 (ii) 一个右非奇异环$R$是一种权利$F\mathrm{一个}$-ring当且仅当$R$是一个完全平方的冯诺依曼正则右自内射环和一个包含其右最大分数环的所有可逆元素的强正则环的直接和。特别是，我们展示了一个右半角（或左半角）环$R$是右非奇异右$F\mathrm{一个}$-ring当且仅当$R$是左非奇异左$F\mathrm{一个}$-戒指。