This is a very controversial subject within the faith. The simple answer as Paul suggests is to judge for yourself. Nevertheless neither is the man without the woman, neither the woman without the man, in the Lord.
There is a New Testament precedent aside from Paul and Timothy’s teachings by Luke.
Serving is also ministry! Teaching is as well.
And it came to pass afterward, that he went throughout every city and village, preaching and shewing the glad tidings of the kingdom of God: and the twelve were with him, And certain women, which had been healed of evil spirits and infirmities, Mary called Magdalene, out of whom went seven devils, And Joanna the wife of Chuza Herod’s steward, and Susanna, and many others, which ministered (served) unto him of their substance.
https://www.biblegateway.com/passage/?search=Luke.8.1,Luke.8.2,Luke.8.3&version=KJV
Paul primarily communicated his own recommendations based on the type of relationships within the congregations.
To the rest I say this (I, not the Lord): If any brother has a wife who is not a believer and she is willing to live with him, he must not divorce her. And if a woman has a husband who is not a believer and he is willing to live with her, she must not divorce him. For the unbelieving husband has been sanctified through his wife, and the unbelieving wife has been sanctified through her believing husband. Otherwise your children would be unclean, but as it is, they are holy. Nevertheless, each person should live as a believer in whatever situation the Lord has assigned to them, just as God has called them. This is the rule I lay down in all the churches.
https://www.biblegateway.com/passage/?search=1Cor.7.12,1Cor.7.13,1Cor.7.14,1Cor.7.17&version=NIV
Timothy had suggestions as well, according to his will as in the NIV passage. While being a student of Paul, Timothy did give a reference to Paul’s teaching of the women being silent and learning from their husband also. However he himself adds; in full submission to her husband and refers to the Genesis account whereas Eve didn’t do the will of her husband Adam.
We will see that Timothy disagrees with women teaching, as in “Desiring to be teachers of the law; understanding neither what they say, nor whereof they affirm.” 1Tim.1.7 to be specific which is biblically accurate.
Now remember their were women in the synagogues.
I also want the women to dress modestly, with decency and propriety, adorning themselves, not with elaborate hairstyles or gold or pearls or expensive clothes, but with good deeds, appropriate for women who profess to worship God. A woman should learn in quietness and full submission. I do not permit a woman to teach or to assume authority over a man; she must be quiet. For Adam was formed first, then Eve. And Adam was not the one deceived; it was the woman who was deceived and became a sinner.
https://www.biblegateway.com/passage/?search=1Tim.2.9,1Tim.2.10,1Tim.2.11,1Tim.2.12,1Tim.2.13,1Tim.2.14&version=NIV
Desiring to be teachers of the law; understanding neither what they say, nor whereof they affirm. I will therefore that men pray every where, lifting up holy hands, without wrath and doubting. In like manner also, that women adorn themselves in modest apparel, with shamefacedness and sobriety; not with broided hair, or gold, or pearls, or costly array; Let the woman learn in silence with all subjection. But I suffer not a woman to teach, nor to usurp authority over the man, but to be in silence. For Adam was first formed, then Eve. And Adam was not deceived, but the woman being deceived was in the transgression.
https://www.biblegateway.com/passage/?search=1Tim.1.7,1Tim.2.8,1Tim.2.9,1Tim.2.11,1Tim.2.12,1Tim.2.13,1Tim.2.14&version=KJV
However, Paul teaches from a relationship perspective and discusses how the women and mens conduct themselves in their household and the church according to the law.
Now I praise you, brethren, that ye remember me in all things, and keep the ordinances, as I delivered them to you. But I would have you know, that the head of every man is Christ; and the head of the woman is the man; and the head of Christ is God. Every man praying or prophesying, having his head covered, dishonoureth his head.
But every woman that prayeth or prophesieth with her head uncovered dishonoureth her head: (Her Husband) for that is even all one as if she were shaven. For a man indeed ought not to cover his head, forasmuch as he is the image and glory of God: but the woman is the glory of the man. For the man is not of the woman: but the woman of the man. Neither was the man created for the woman; but the woman for the man. For this cause ought the woman to have power on her head because of the angels.
Nevertheless neither is the man without the woman, neither the woman without the man, in the Lord. Judge in yourselves: is it comely that a woman pray unto God uncovered?
https://www.biblegateway.com/passage/?search=1Cor.11.2,1Cor.11.3,1Cor.11.4,1Cor.11.5,1Cor.11.7,1Cor.11.8,1Cor.11.9,1Cor.11.10,1Cor.11.11,1Cor.11.13&version=KJV
Paul is first asking the men how every one of them seem to have something to express to the others when they gather. He then reminds them of the correct order of things mentioned in the Law when they come together.
How is it then, brethren? When ye come together, every one of you hath a psalm, hath a doctrine, hath a tongue, hath a revelation, hath an interpretation. Let all things be done unto edifying. If any man speak in an unknown tongue, let it be by two, or at the most by three, and that by course; and let one interpret. But if there be no interpreter, let him keep silence in the church; and let him speak to himself, and to God.
He tells the Brethren if any speak in an unknown tongue (while gathered within the group) to let one (brethren) interpret. And if there is NO interpreter, let HIM (the unknown tongue speaker without an interpreter) keep silent in the church and speak to himself and to God (in silence while gathered with others -church body)
Let YOUR women keep silence in the churches: (wives/daughters of the brethren) for it is not permitted unto them to speak; but they are commanded to be under obedience as also saith the law. (The husbands cover the wives vows in the law).
And if they will learn any thing, let them ask their husbands at home: for it is a shame for women to speak in the church. What? came the word of God out from you? or came it unto you only? If any man think himself to be a prophet, or spiritual, let him acknowledge that the things that I write unto you are the commandments of the Lord.
https://www.biblegateway.com/passage/?search=1Cor.14.26,1Cor.14.27,1Cor.14.28,1Cor.14.34,1Cor.14.35,1Cor.14.36,1Cor.14.37&version=KJV
For context, Moses had given the commandment which Paul was referring to in the Law.
And Moses spake unto the heads of the tribes concerning the children of Israel, saying, This is the thing which the Lord hath commanded. If a man vow a vow unto the Lord, or swear an oath to bind his soul with a bond; he shall not break his word, he shall do according to all that proceedeth out of his mouth. If a woman also vow a vow unto the Lord, and bind herself by a bond, being in her father’s house in her youth; And if she had at all an husband, when she vowed, or uttered ought out of her lips, wherewith she bound her soul; Every vow, and every binding oath to afflict the soul, her husband may establish it, or her husband may make it void. But if her husband altogether hold his peace at her from day to day; then he establisheth all her vows, or all her bonds, which are upon her: he confirmeth them, because he held his peace at her in the day that he heard them.
But if he shall any ways make them void after that he hath heard them; then he shall bear her iniquity. These are the statutes, which the Lord commanded Moses, between a man and his wife, between the father and his daughter, being yet in her youth in her father’s house.
https://www.biblegateway.com/passage/?search=Num.30.1,Num.30.2,Num.30.3,Num.30.6,Num.30.13,Num.30.14,Num.30.15,Num.30.16&version=KJV
For the end of the commandment is love out of a pure heart, and of a good conscience, and of faith unfeigned. From the which things some have erred, and have turned unto vain jangling. They would be doctors of the Law, and yet understand not what they speak, neither whereof they affirm.
https://www.biblegateway.com/passage/?search=1Tim.1.5,1Tim.1.6,1Tim.1.7&version=KJV
As far as women ruling, there is biblical precedent such as Deborah the Judge. Deborah, the wife of Lappidoth, was a prophet who was judging Israel at that time. She would sit under the Palm of Deborah, between Ramah and Bethel in the hill country of Ephraim, and the Israelites would go to her for judgment.
https://www.biblegateway.com/passage/?search=Judg.4.4,Judg.4.5&version=NLT
and in various passages including that of Phillip’s encounter with the Ethiopian eunuch, which led back to a Kushite queen. https://zondervanacademic.com/blog/whats-a-kandake-acts-827-mondays-with-mounce-233